confidence intervals & sample size chapter 7. introduction one aspect of inferential statistics...
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Confidence Intervals & Sample SizeChapter 7
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Introduction• One aspect of inferential statistics is estimation, which is the
process of estimating the value of a parameter from information obtained from a sample
• One important question exists:• How large should the sample be in order to make an accurate
estimate?
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7.1 – Confidence Intervals for the Mean When σ is Known• Point estimate:• Specific numerical value estimate of a parameter• The best point estimate of a population mean μ is the sample
mean
Three Properties of a Good Estimator1. The estimator should be an unbiased estimator2. The estimator should be consistent estimator3. The estimator should be a relatively efficient estimator
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Interval Estimates• Interval estimate• Interval or a range of values used to estimate the parameter• This estimate may or may not contain the value of the parameter
being estimated
• Confidence level• Probability that the interval estimate will contain the parameter,
assuming that a large number of samples are selected and that the estimation process on the same parameter is repeated
• Confidence interval• Determined by using data obtained from a sample and by using
the specific confidence level of the estimate
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Confidence Interval Formula• Three common confidence levels exist: 90, 95, and 99%
Formula for the Confidence Interval of the Mean
• Maximum error of the mean:• Maximum likely difference between the point estimate of a
parameter and the actual value of the parameter• is the maximum error of the mean (also called margin of error)
• Rounding rule:• Round off to one more decimal place than original data
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Examples• 7 – 1• A researcher wishes to estimate the number of days it takes an
automobile dealer to sell a Chevrolet Aveo. A sample of 50 cars had a mean time on the dealer’s lot of 54 days. Assume the population standard deviation to be 6.0 days. Find the best point estimate of the population mean and the 95% confidence interval of the population mean.
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Example• 7 – 2• A survey of 30 adults found that the mean age of a person’s
primary vehicle is 5.6 years. Assuming the standard deviation of the population is 0.8 year, find the best point estimate of the population mean and the 99% confidence interval of the population mean.
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Examples• 7 – 3• The following data represent a sample of the assets (in millions of
dollars) of 30 credit unions in southwestern Pennsylvania. Find the 90% confidence interval of the mean. (use calculators)
12.23 16.56 4.39 2.8913.19 73.25 11.59 8.747.92 40.22 5.01 2.271.24 9.16 1.91 6.693.17 4.78 2.42 1.4712.77 2.17 1.42 14.641.06 18.13 16.85 21.5812.24 2.76
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Sample Size• One important question exists…• How large a sample is necessary to make an accurate estimate?
Formula for the Minimum Sample Size Needed for an Interval Estimate of the Population Mean
• Where E is the maximum error of the estimate. If necessary, round the answer up to obtain a whole number.
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Examples• 7 – 4• A scientist wishes to estimate the average depth of a river. He
wants to be 99% confident that the estimate is accurate within 2 feet. From a previous study, the standard deviation of the depths measured was 4.38
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7.2 – Confidence Intervals when σ is Unknown• Most of the time the population standard deviation (σ) is not
known
• It is estimated using the sample standard deviation (s)
• Values are taken from the t distribution
• The t distribution was formulated in 1908 by an Irish brewing employee named W.S. Gosset
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Characteristics of thet Distribution• The t distribution shares some characteristics of the normal distribution
and differs it in others.
The t distribution is similar to the normal distribution in these ways:1. Bell-shaped2. Symmetric about the mean3. Mean, median, and mode are equal to 0 and are located at the center of
the distribution4. Curve never touches the x axis
The t distribution differs from the normal distribution in these ways:5. Variance is greater than 16. t distribution is a family of curves based on concept of degrees of
freedom which is related to sample size7. As sample size increases, t distribution approaches normal distribution
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Degrees of Freedom• Degrees of freedom• Number of values that are free to vary after a sample statistic has
been computed• Tell the researcher which specific curve to use• The symbol d.f. is used for degrees of freedom• Degrees of freedom for a confidence interval for the mean are
found by subtracting 1 from the sample size: d.f. = n – 1
• When d.f. is greater than 30, it may fall between two table values
• Always round down to the nearest table value• Example: 68 is closer to 70, but round down to table value of 65
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Formula
Formula for a Specific Confidence Interval for the Mean When σ is Unknown and n < 30
• The degrees of freedom are n – 1
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Examples• 7 – 5• Find the ta/2 value for a 95% confidence interval when the sample
size is 22
• 7 – 6• Ten randomly selected people were asked how long they slept at
night. The mean time was 7.1 hours, and the standard deviation was 0.78 hour. Find the 95% confidence interval of the mean time. Assume the variable is normally distributed.
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Example 7 – 7• The data represent a sample of the number of home fires
started by candles for the past several years. Find the 99% confidence interval for the mean number of home fires started by candles each year.
5460 5900 6090 6310 7160 8440 9930
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7.3 – Confidence Intervals and Sample Size for Proportions• Proportion• Represents a part of a whole expressed as fraction, decimal, or
percentage• Proportions can also represent probabilities
Symbols Used in Proportion Notation
For a sample proportion:and or
where X = number of sample units that posses the characteristics of interest and n = sample size
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Example 7 – 8• In a recent survey of 150 households, 54 had central air
conditioning. Find p hat and q hat where p hat is the proportion of households that have central air conditioning.
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Formula• Maximum error of the estimate is
• Confidence intervals about proportions must meet the criteria that
Formula for a Specific Confidence Interval for a Proportion
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Examples• 7 – 9• A sample of 500 nursing applications included 60 from men. Find
the 90% confidence interval of the true proportion of men who applied to the nursing program.
• 7 – 10• A survey of 1721 people found that 15.9% of individuals purchase
religious books at a Christian bookstore. Find the 95% confidence interval of the true proportion of people who purchase their religious books at a Christian bookstore.
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Sample Size for Proportions• To find the sample size needed to determine a confidence
interval about a proportion use:
Formula for Minimum Sample Size Needed for Interval Estimate of a Population Proportion
• If necessary, round up to obtain a whole number
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Examples• 7 – 11• A researcher wishes to estimate, with 95% confidence, the
proportion of people who own a home computer. A previous study shows that 40% of those interviewed had a computer at home. The researcher wishes to be accurate within 2% of the true proportion. Find the minimum same size necessary.
• 7 – 12• The same researcher wishes to estimate the proportion of
executives who own a car phone. She wants to be 90% confident and be accurate within 5% of the true proportion. Find the minimum sample size necessary.
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7.4 – Confidence Intervals for Variances & Stand. Deviations• In statistics, the variance and standard deviation of a variable
are as important as the mean• For example:• Products that fit together (such as pipes) are manufactured so
that variations in diameters are as small as possible
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Chi-Square Distribution• To calculate these confidence intervals, a new statistical
distribution is needed
• Chi-square distribution• Similar to the t variable in that its distribution is a family of curves
based of the number of degrees of freedom• Symbol for chi-square is • Chi-square variable cannot be negative• Area under the chi-square distribution is 1.00 or 100%
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Example 7 – 13• Find the values of and for a 90% confidence interval when n =
25
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Formulas• Formulas for confidence intervals where d.f. = n – 1
Formula for the Confidence Interval for a Variance
Formula for the Confidence Interval for a Standard Deviation
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Examples• 7 – 14• Find the 95% confidence interval for the variance and standard
deviation of the nicotine content of cigarettes manufactured if a sample of 20 cigarettes has a standard deviation of 1.6 milligrams.
• 7 – 15• Find the 90% confidence interval for the variance and standard
deviation for the price in dollars of an adult single-day ski lift ticket. The data represent a selected sample of nationwide ski resorts. Assume the variable is normally distributed.
59 54 53 52 5139 49 46 49 48