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Slides by

SpirosVelianiti

sCSUS

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Review ObjectiveReview Objective

The objective of this section is to ensure that you The objective of this section is to ensure that you have the necessary foundation in statistics so that you have the necessary foundation in statistics so that you can maximize your learning in data analysis. can maximize your learning in data analysis.

Hopefully, much of this material will be review.Hopefully, much of this material will be review. Instead of repeating Statistics 1, Instead of repeating Statistics 1, the pre-requisite for the pre-requisite for

this course, we discuss some major topics with the this course, we discuss some major topics with the intention that you will focus on concepts intention that you will focus on concepts and not be and not be overly concerned with details. overly concerned with details.

In other words, as we “review” try to think of the In other words, as we “review” try to think of the overall picture!overall picture!

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Stat 1 Review Summary SlideStat 1 Review Summary Slide

Statistic vs. ParameterStatistic vs. Parameter Mean and VarianceMean and Variance Sampling DistributionSampling Distribution Normal DistributionNormal Distribution Student’s t-DistributionStudent’s t-Distribution Confidence IntervalsConfidence Intervals Hypothesis TestingHypothesis Testing

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Statistic vs. ParameterStatistic vs. Parameter In order for managers to make good decisions, they need information. In order for managers to make good decisions, they need information.

Information is derived by summarizing data (often obtained via samples) since Information is derived by summarizing data (often obtained via samples) since data, in its original form, is hard to interpret. This is where statistics come into data, in its original form, is hard to interpret. This is where statistics come into play -- play -- a statistic is nothing more than a quantitative value calculated from a a statistic is nothing more than a quantitative value calculated from a samplesample..

There are many different statistics that can be calculated from a sample. Since There are many different statistics that can be calculated from a sample. Since we are interested in using statistics to make decisions there usually are only a we are interested in using statistics to make decisions there usually are only a few statistics we are interested in using. These useful statistics estimate few statistics we are interested in using. These useful statistics estimate characteristics of the population, which when quantified are called characteristics of the population, which when quantified are called parameters. parameters.

Some of the most common parameters are μ (population mean), σ Some of the most common parameters are μ (population mean), σ 22 (population (population variance), σ (population standard deviation), and Ρ (population proportion).variance), σ (population standard deviation), and Ρ (population proportion).

The key point here is that managers must make decisions based upon their The key point here is that managers must make decisions based upon their perceived values of parameters. Usually the values of the parameters are perceived values of parameters. Usually the values of the parameters are unknown. Thus, managers must rely on data from the population (sample), unknown. Thus, managers must rely on data from the population (sample), which is summarized (statistics), in order to estimate the parameters. The which is summarized (statistics), in order to estimate the parameters. The corresponding statistics used to estimate the parameters listed above (μ , σcorresponding statistics used to estimate the parameters listed above (μ , σ 2 2, , σ , and Ρ ) are called ˆσ , and Ρ ) are called ˆx (sample mean), sx (sample mean), s22 (sample variance), s (sample (sample variance), s (sample standard deviation), standard deviation), and ˆand ˆp (sample proportion).p (sample proportion).

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Mean and VarianceMean and Variance Two very important parameters which managers focus on frequently are Two very important parameters which managers focus on frequently are

the the meanmean and and variancevariance.. The mean, which is frequently referred to as “the average,” provides a The mean, which is frequently referred to as “the average,” provides a

measure of the central locationmeasure of the central location The The variance describes the amount of dispersion variance describes the amount of dispersion within the within the

population. population. TThe square root of the variance is called a he square root of the variance is called a standard standard deviationdeviation. . In finance, the standard deviation of the stock returns is called In finance, the standard deviation of the stock returns is called volatility.volatility.

For example, consider a portfolio of stocks. When discussing the rate of For example, consider a portfolio of stocks. When discussing the rate of return from such a portfolio, and knowing that the rate of return will vary return from such a portfolio, and knowing that the rate of return will vary from time period to time period one may wish to know the average rate of from time period to time period one may wish to know the average rate of return (mean) and how much variation there is in the returns. The rate of return (mean) and how much variation there is in the returns. The rate of return is calculated as follows:return is calculated as follows:

return= (New Price - Old Price)/Old Pricereturn= (New Price - Old Price)/Old Price The The medianmedian is another measure of central location and is the value in the is another measure of central location and is the value in the

middle when the data are arranged in ascending order. middle when the data are arranged in ascending order. The The modemode is a third measure of central location and is the value that is a third measure of central location and is the value that

occurs the most often in the data.occurs the most often in the data.

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ExercisesExercises

1. Explain the difference between mean and median. 1. Explain the difference between mean and median. Why does the media report median more often than Why does the media report median more often than the mean for family income, housing price, rents, the mean for family income, housing price, rents, etc.?etc.?

2. Explain why investors might be interested in the 2. Explain why investors might be interested in the mean and variance of stock market return.mean and variance of stock market return.

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Sampling DistributionSampling Distribution

In order to understand statistics and not just “plug” numbers into In order to understand statistics and not just “plug” numbers into formulas, one needs to understand the concept of a sampling formulas, one needs to understand the concept of a sampling distribution. In particular, one needs to know that distribution. In particular, one needs to know that every statistic every statistic has a sampling distribution, which shows every possible has a sampling distribution, which shows every possible value the statistic can take on and the corresponding value the statistic can take on and the corresponding probability of occurrenceprobability of occurrence..

What does this mean in simple terms? Consider a situation where What does this mean in simple terms? Consider a situation where you wish to calculate the mean age of all students at CSUS. If you you wish to calculate the mean age of all students at CSUS. If you take a random sample of size 25, you will get one value for the take a random sample of size 25, you will get one value for the sample mean (average) which may or may not be the same as the sample mean (average) which may or may not be the same as the sample mean from the first sample. Suppose you get another sample mean from the first sample. Suppose you get another random sample of size 25, will you get the same sample mean? random sample of size 25, will you get the same sample mean? What if you take many samples, each of size 25, and you graph What if you take many samples, each of size 25, and you graph the distribution of sample means. What would such a graph show? the distribution of sample means. What would such a graph show? The answer is that it will show the distribution of sample means, The answer is that it will show the distribution of sample means, from which from which probabilistic statements about the population mean probabilistic statements about the population mean can be made.can be made.

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Normal DistributionNormal Distribution For the situation described in the previous slide, the distribution of the For the situation described in the previous slide, the distribution of the

sample mean will follow a normal distribution. What is a sample mean will follow a normal distribution. What is a normal normal distribution? The normal distribution has the following distribution? The normal distribution has the following attributes attributes (suppose the random variable Χ follows a normal distribution):(suppose the random variable Χ follows a normal distribution):

1.1. It is bell-shapedIt is bell-shaped

2.2. It is symmetrical about the mean. It is symmetrical about the mean.

3.3. It depends on two parameters - the It depends on two parameters - the mean ( μ ) and variance ( σmean ( μ ) and variance ( σ22)) From a manager’s perspective it is very important to know that with From a manager’s perspective it is very important to know that with

normal distributions approximately:normal distributions approximately:

68% of all observations fall within 1 standard deviations of the mean: 68% of all observations fall within 1 standard deviations of the mean:

ΡΡrob(rob(μ −σ ≤ Χ ≤μ +σ)≈ 0.68 .μ −σ ≤ Χ ≤μ +σ)≈ 0.68 .

95% of all observations fall within 2 standard deviations of the mean:95% of all observations fall within 2 standard deviations of the mean:

Ρrob(μ −2σ ≤Χ≤μ +2σ)≈0.95.Ρrob(μ −2σ ≤Χ≤μ +2σ)≈0.95.

99.7% of all observations fall within 3 standard deviations of the mean:99.7% of all observations fall within 3 standard deviations of the mean:

Ρrob(μ −3σ ≤Χ≤μ +3σ)≈0.997 .Ρrob(μ −3σ ≤Χ≤μ +3σ)≈0.997 . When μ =0 and σ =1 , we have the so-called When μ =0 and σ =1 , we have the so-called standard normal standard normal

distribution, usually distribution, usually denoted by Ζ . It is also called the denoted by Ζ . It is also called the Z-score.Z-score. Look at Look at http://www.statsoft.com/textbook/sttable.html#z

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Central Limit TheoremCentral Limit Theorem

A very important theorem from your Stat 1 course is A very important theorem from your Stat 1 course is called the Central Limit Theorem.called the Central Limit Theorem.

The Central Limit Theorem states that The Central Limit Theorem states that the distribution the distribution of sample means is approximately normal of sample means is approximately normal provided that provided that the the sample size is large enoughsample size is large enough. “Central” here means . “Central” here means important.important.

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Chi-Square (χChi-Square (χ22) Distribution and the ) Distribution and the F F DistributionDistribution

The sample varianceThe sample variance

is the point estimator of the population variance σis the point estimator of the population variance σ22 . . To make inference about the population variance σTo make inference about the population variance σ2 2 using the using the

sample variance sample variance ss22 , the sampling distribution of , the sampling distribution of (n−1)s(n−1)s22/σ/σ22 applies applies and and it follows a chi-square distribution with it follows a chi-square distribution with n −1 degrees of n −1 degrees of freedom, where n is the sample size of freedom, where n is the sample size of a simple random sample a simple random sample from a normal population. from a normal population.

The Chi-square distribution is the sum of squared independent The Chi-square distribution is the sum of squared independent standard normal variables Σ with standard normal variables Σ with v degrees of freedom. v degrees of freedom.

The The F distribution is a ratio of two Chi-Square distributions.F distribution is a ratio of two Chi-Square distributions. You will see both distributions later in You will see both distributions later in the course.the course.

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How to Look up the Chi Square Distribution How to Look up the Chi Square Distribution tabletable

The The Chi-squareChi-square distribution's shape is determined by its degrees of distribution's shape is determined by its degrees of freedom. freedom.

As shown in the illustration below, the values inside this table are critical As shown in the illustration below, the values inside this table are critical values of the Chi-square distribution with the corresponding degrees of values of the Chi-square distribution with the corresponding degrees of freedom. To determine the value from a Chi-square distribution (with a freedom. To determine the value from a Chi-square distribution (with a specific degree of freedom) which has a given area above it, go to the specific degree of freedom) which has a given area above it, go to the given area column and the desired degree of freedom row. For example, given area column and the desired degree of freedom row. For example, the .25 critical value for a Chi-square with 4 degrees of freedom is the .25 critical value for a Chi-square with 4 degrees of freedom is 5.38527. This means that the area to the right of 5.38527 in a Chi-square 5.38527. This means that the area to the right of 5.38527 in a Chi-square distribution with 4 degrees of freedom is .25.distribution with 4 degrees of freedom is .25.

Right tail areas for the Chi-square Distribution Right tail areas for the Chi-square Distribution

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Student’s t-DistributionStudent’s t-Distribution

When the population standard deviation σ in the Z-score above is When the population standard deviation σ in the Z-score above is replaced by the sample standard deviation replaced by the sample standard deviation s , this score follows s , this score follows the the t-distribtuion. t-distribtuion.

It is bell-shaped just like the normal It is bell-shaped just like the normal distribution with mean distribution with mean 0. It is more spread out than the standard normal distribution. In 0. It is more spread out than the standard normal distribution. In other words, the two tails are heavier than those of the standard other words, the two tails are heavier than those of the standard normal distribution. normal distribution.

The The t-distribution t-distribution has a parameter called has a parameter called degrees of degrees of freedom (df). In most applications, it is a function of the freedom (df). In most applications, it is a function of the sample size but the specific formula depends on the problem. sample size but the specific formula depends on the problem.

The The t-distribution is in fact a ratio t-distribution is in fact a ratio of a standard normal of a standard normal distribution to a chi-square distribution. distribution to a chi-square distribution.

When degrees of freedom increase, the When degrees of freedom increase, the t-distribution t-distribution approaches the standard normal distribution.approaches the standard normal distribution.

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How to Look up the t Distribution tableHow to Look up the t Distribution table

The Shape of the Student's t distribution is determined by the The Shape of the Student's t distribution is determined by the degrees of freedom. Its shape changes as the degrees of freedom degrees of freedom. Its shape changes as the degrees of freedom increases. As indicated by the chart below, the areas given at the top increases. As indicated by the chart below, the areas given at the top of this table are the right tail areas for the t-value inside the table. To of this table are the right tail areas for the t-value inside the table. To determine the 0.05 critical value from the t-distribution with 6 determine the 0.05 critical value from the t-distribution with 6 degrees of freedom, look in the 0.05 column at the 6 row: tdegrees of freedom, look in the 0.05 column at the 6 row: t(.05,6)(.05,6) = = 1.943180. 1.943180.

t table with right tail probabilities:t table with right tail probabilities:

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Confidence IntervalsConfidence Intervals Constructing a confidence interval estimate of the Constructing a confidence interval estimate of the

unknown value of a population parameter is one of the unknown value of a population parameter is one of the most common statistical inference procedures. most common statistical inference procedures.

A confidence interval is an interval of values computed A confidence interval is an interval of values computed from sample data that is likely to include the true from sample data that is likely to include the true population value. population value.

The term The term confidence level is the chance that this confidence level is the chance that this confidence interval actually contains the true confidence interval actually contains the true population valuepopulation value..

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Confidence Interval ExampleConfidence Interval Example Suppose you wish to make an inference about the average income for all Suppose you wish to make an inference about the average income for all

students at Sacramento State (population mean μ, a parameter). From a students at Sacramento State (population mean μ, a parameter). From a sample of 45 Sacramento State students, one can come up with a sample of 45 Sacramento State students, one can come up with a point point estimate (a sample statistic used to estimate a population estimate (a sample statistic used to estimate a population parameter), such as $24,000. But what does this mean? parameter), such as $24,000. But what does this mean? A point estimate A point estimate does not take into account the accuracy of the calculated statistic. does not take into account the accuracy of the calculated statistic. We also We also need to know the need to know the variationvariation of our estimate. We are not absolutely certain of our estimate. We are not absolutely certain that the mean income for Sacramento State students is $24,000 since this that the mean income for Sacramento State students is $24,000 since this sample mean is only an estimate of the population mean. If we collect sample mean is only an estimate of the population mean. If we collect another sample of 45 Sacramento State students, we would have another another sample of 45 Sacramento State students, we would have another estimate of the mean. Thus, different samples yield different estimates of estimate of the mean. Thus, different samples yield different estimates of the mean for the same population. the mean for the same population. How close these sample means are to How close these sample means are to one another determines the variation of the estimate of the population one another determines the variation of the estimate of the population mean.mean.

A statistic that measures the variation of our estimate is the A statistic that measures the variation of our estimate is the standard error standard error of the meanof the mean. It is . It is different from the sample standard deviation ( different from the sample standard deviation ( s ) s ) because the sample standard deviation reveals because the sample standard deviation reveals the variation of our data. the variation of our data.

The standard error of the mean reveals the variation of our sample mean. The standard error of the mean reveals the variation of our sample mean. The standard error of the mean is a measure of how much error we can The standard error of the mean is a measure of how much error we can expect when we use the sample mean to predict the population mean. The expect when we use the sample mean to predict the population mean. The smaller the standard error is, the more accurate our sample estimate is.smaller the standard error is, the more accurate our sample estimate is.

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Confidence Interval Example (cont)Confidence Interval Example (cont) In order to provide additional information, one needs to provide a In order to provide additional information, one needs to provide a

confidence interval. confidence interval. A A confidence interval is a range of values that one believes confidence interval is a range of values that one believes

to contain the population parameter of to contain the population parameter of interest and places an interest and places an upper and lower bound around a sample statistic. upper and lower bound around a sample statistic.

To construct a confidence interval, we need to choose a To construct a confidence interval, we need to choose a significance level. A 95% (=1-5% where 5% is the significance level. A 95% (=1-5% where 5% is the level of level of significance or α ) confidence interval is often used to significance or α ) confidence interval is often used to assess the variability of the sample assess the variability of the sample mean. A 95% confidence mean. A 95% confidence interval for the mean student income means we are 95% confident interval for the mean student income means we are 95% confident the interval contains the mean income for Sacramento State the interval contains the mean income for Sacramento State students. We want to be as confident as possible. However, if we students. We want to be as confident as possible. However, if we increase the confidence level, the width of our confidence interval increase the confidence level, the width of our confidence interval increases. As the width of the interval increases, it becomes less increases. As the width of the interval increases, it becomes less useful. What is the difference between the following 95% useful. What is the difference between the following 95% confidence intervals for the population mean?confidence intervals for the population mean?

[23000, 24500] and [12000, 36000][23000, 24500] and [12000, 36000]

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Confidence Interval Hands On Example Confidence Interval Hands On Example (Pg 10).(Pg 10).

The following is a sample of regular gasoline price in Sacramento The following is a sample of regular gasoline price in Sacramento on May 22, 2006 found at www. automotive.com:on May 22, 2006 found at www. automotive.com:

3.299 3.189 3.269 3.279 3.299 3.249 3.319 3.239 3.2193.299 3.189 3.269 3.279 3.299 3.249 3.319 3.239 3.219

3.249 3.299 3.239 3.319 3.359 3.169 3.299 3.299 3.2393.249 3.299 3.239 3.319 3.359 3.169 3.299 3.299 3.239 Find the 95% confidence interval for the population mean.Find the 95% confidence interval for the population mean. Given the small sample size of 18, the Given the small sample size of 18, the t-distribution should be t-distribution should be

used. To find the 95% confidence used. To find the 95% confidence interval for the population interval for the population mean using this sample, you need to mean using this sample, you need to x , s , n , and tx , s , n , and tα/2α/2 . .

Then α = 0.05 (from 1-0.95), Then α = 0.05 (from 1-0.95), n = 18 , n = 18 , ^̂x = 3.268 , s = 0.0486 , n = x = 3.268 , s = 0.0486 , n = 18, degrees of freedom=18-1=17, 18, degrees of freedom=18-1=17, and and tt0.05/2 0.05/2 = 2.11 . Plug these = 2.11 . Plug these values into the formula below.values into the formula below.

Thus, we are 95% confident that the true mean of regular gas Thus, we are 95% confident that the true mean of regular gas price in Sacramento is between 3.244 and 3.293. The formal price in Sacramento is between 3.244 and 3.293. The formal interpretation is that in repeated sampling, the interval will contain interpretation is that in repeated sampling, the interval will contain the true mean of the population from which the data come 95% of the true mean of the population from which the data come 95% of the time.the time.

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Hypothesis TestingHypothesis Testing Consider the following scenario:Consider the following scenario:

I invite you to play a game where I pull a coin out and toss it. If it I invite you to play a game where I pull a coin out and toss it. If it comes up heads you pay me $1. Would you be willing to play? To comes up heads you pay me $1. Would you be willing to play? To decide whether to play or not, many people would like to know if decide whether to play or not, many people would like to know if the coin is fair. To determine if you think the coin is fair (a the coin is fair. To determine if you think the coin is fair (a hypothesis) or not (alternative hypothesis) you might take the coin hypothesis) or not (alternative hypothesis) you might take the coin and toss it a number of times, recording the outcomes (data and toss it a number of times, recording the outcomes (data collection). Suppose you observe the following sequence of collection). Suppose you observe the following sequence of outcomes, here H represents a head and T represents a tail -outcomes, here H represents a head and T represents a tail -

H H H H H H H H T H H H H H H T H H H H H HH H H H H H H H T H H H H H H T H H H H H H

What would be your conclusion? Why?What would be your conclusion? Why? Most people look at the observations and notice the large number Most people look at the observations and notice the large number

of heads (statistic) and conclude that they think the coin is not fair of heads (statistic) and conclude that they think the coin is not fair because the probability of getting 20 heads out of 22 tosses is very because the probability of getting 20 heads out of 22 tosses is very small, if the coin is fair (sampling distribution). It did happen; small, if the coin is fair (sampling distribution). It did happen; hence one rejects the idea of a fair coin and consequently does not hence one rejects the idea of a fair coin and consequently does not wish to participate in the game.wish to participate in the game.

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Hypothesis Testing StepsHypothesis Testing Steps

11. State hypothesis. State hypothesis

2. Collect data2. Collect data

3. Calculate test statistic3. Calculate test statistic

4. Determine likelihood of outcome, if null hypothesis is true4. Determine likelihood of outcome, if null hypothesis is true

5. If the likelihood is small, then reject the null hypothesis5. If the likelihood is small, then reject the null hypothesis The one question that needs to be answered is “what is small?” To The one question that needs to be answered is “what is small?” To

quantify what quantify what small is small is one needs to understand the concept of a one needs to understand the concept of a Type I error. As you may recall from your Stat 1 course, there are Type I error. As you may recall from your Stat 1 course, there are the null ( 0 the null ( 0 H ) and alternative ( 1 H ) hypotheses. Either one of H ) and alternative ( 1 H ) hypotheses. Either one of them is true. Our them is true. Our test procedure should lead to accept 0 test procedure should lead to accept 0 H when 0 H when 0 H is true and reject 0 H if 1 H is true, H is true and reject 0 H if 1 H is true, ideally. ideally. However, this not However, this not always the case and errors could be made. always the case and errors could be made. Type I error is made Type I error is made if a true 0 if a true 0 H H is rejected. is rejected. Type II error is made if a false 0 Type II error is made if a false 0 H is H is accepted. This is summarized below:accepted. This is summarized below:

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P-ValuesP-Values In order to simplify the decision-making process for In order to simplify the decision-making process for

hypothesis testing, hypothesis testing, p-values are frequently p-values are frequently reported reported when the analysis is performed on the computer. In when the analysis is performed on the computer. In particular a particular a p-value refers to where in the sampling p-value refers to where in the sampling distribution the test statistic resides. distribution the test statistic resides. Hence the decision Hence the decision rules managers can use are:rules managers can use are:

• • If the p-value is < alpha, then reject HoIf the p-value is < alpha, then reject Ho

• • If the p-value is >=alpha, then do not reject Ho.If the p-value is >=alpha, then do not reject Ho. The p-value may be defined The p-value may be defined as as the probability of the probability of

obtaining a test statistic equal to or more extreme than obtaining a test statistic equal to or more extreme than the result obtained from the sample data, given the null the result obtained from the sample data, given the null hypothesis H0 is really true.hypothesis H0 is really true.

Can you explain this concept with an example?Can you explain this concept with an example?

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P-Values ExampleP-Values Example

In 1991 the average interest rate charged by U.S. credit In 1991 the average interest rate charged by U.S. credit card issuers was 18.8%. Since that time, there has been a card issuers was 18.8%. Since that time, there has been a proliferation of new credit cards affiliated with retail proliferation of new credit cards affiliated with retail stores, oil companies, alumni associations, professional stores, oil companies, alumni associations, professional sports teams, and so on. A financial officer wishes to study sports teams, and so on. A financial officer wishes to study whether the increased competition in the credit card whether the increased competition in the credit card business has reduced interest rates. To do this, the officer business has reduced interest rates. To do this, the officer will test a hypothesis about the current mean interest, μ , will test a hypothesis about the current mean interest, μ , charged by U.S. credit card issuers. The null hypothesis to charged by U.S. credit card issuers. The null hypothesis to be tested is be tested is HH00 :μ ≥18.8% , and the alternative hypothesis :μ ≥18.8% , and the alternative hypothesis is His H11:μ <18.8%. If H:μ <18.8%. If H00 can be rejected in favor of H can be rejected in favor of H1 1 at the at the 0.05 level of significance, the officer 0.05 level of significance, the officer will conclude that the will conclude that the current mean interest rate is less than the 18.8% mean current mean interest rate is less than the 18.8% mean interest rate charged in 1991.interest rate charged in 1991.

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Exercise 1Exercise 1 The interest rates in percentage for the 15 sampled The interest rates in percentage for the 15 sampled

cards are: 15.6, 17.8, 14.6, 17.3, 18.7, 15.3, 16.4, 18.4, cards are: 15.6, 17.8, 14.6, 17.3, 18.7, 15.3, 16.4, 18.4, 17.6, 14.0, 19.2, 15.8, 18.1, 16.6, 17.017.6, 14.0, 19.2, 15.8, 18.1, 16.6, 17.0

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Exercise 2Exercise 2

Use Use StatGraphics to test the following StatGraphics to test the following hypothesis for both SP500 and NASDAQ (data hypothesis for both SP500 and NASDAQ (data file: file: sp500nas.xls):sp500nas.xls):

H0: Daily return <= 0H0: Daily return <= 0

H1: Daily return > 0H1: Daily return > 0

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Stat 1 Review Summary SlideStat 1 Review Summary Slide

Statistic vs. ParameterStatistic vs. Parameter Mean and VarianceMean and Variance Sampling DistributionSampling Distribution Normal DistributionNormal Distribution Student’s t-DistributionStudent’s t-Distribution Confidence IntervalsConfidence Intervals Hypothesis TestingHypothesis Testing