Chapter 17/18 Hypothesis Testing

•State the hypotheses. •Formulate an analysis plan. •Analyze sample data. •Interpret the results.

Example

• Portable Phones: A manufacturer claims that a new design for a portable phone has increased the range to 150 feet, allowing many customers to use the phone through out their homes and yards. An independent testing laboratory found that a random sample of 46 of these phones worked over an average distance of 142 feet with a standard deviation of 12 feet. Is there evidence that the manufacturer’s claim is false?

Chapter 17/18 Hypothesis Testing

• State the hypotheses. Every hypothesis test requires the analyst to state a – null hypothesis and

– An alternative hypothesis.

– The hypotheses are stated in such a way that they are mutually exclusive.

– That is, if one is true, the other must be false; and vice versa.

Chapter 17/18 Hypothesis Testing

• Null hypothesis. The null hypothesis, denoted by Ho, is usually the hypothesis that sample observations result purely from chance.

– Statement that you try to find evidence against– Statement of no difference– Statement of no effect

• Alternative hypothesis. The alternative hypothesis, denoted by H1 or Ha, is the hypothesis that sample observations are influenced by some non-random cause.

– Statement that you try to find evidence for– Statement of difference– Statement of effect

Chapter 17/18 Hypothesis Testing

• Formulate an analysis plan. The analysis plan describes how to use sample data to accept or reject the null hypothesis. It should specify the following elements.

– Significance level α. Often, researchers choose significance levels equal to 0.01, 0.05, or 0.10; but any value between 0 and 1 can be used.

– Test method. Typically, the test method involves a test statistic and a sampling distribution. Computed from sample data, the test statistic might be

• a mean score, proportion, • difference between means, • difference between proportions, • z-score, t-score, chi-square, etc.

– P-Value: Given a test statistic and its sampling distribution, a researcher can assess probabilities associated with the test statistic.

– If the test statistic probability is less than the significance level, the null hypothesis is rejected

Chapter 17/18 Hypothesis Testing

• Analyze sample data. Using sample data, perform computations called for in the analysis plan.

Test statistic. When the null hypothesis involves a mean or proportion, use either of the following equations to compute the test statistic.

• Test statistic = (Statistic - Parameter) / (Standard deviation of statistic)

• Test statistic = (Statistic - Parameter) / (Standard error of statistic)

• P-value. The P-value is the probability of observing a sample statistic as extreme as the test statistic, assuming the null hypothesis is true.

Chapter 17/18 Hypothesis Testing

• Interpret the results. If the sample findings are unlikely, given the null hypothesis, the researcher rejects the null hypothesis.

• Typically, this involves comparing the P-value to the significance level, and

• rejecting the null hypothesis when the P-value is less than the significance level.

Chapter 17/18 Hypothesis Testing

Chapter 17/18 Hypothesis Testing

Chapter 17/18 Hypothesis Testing

Type I and II Errors

Type I and II Errors

Type I and II Errors

Example 1

• In the last mayoral election in a large city, 47% of the adults over the age of 65 voted Republican. A researcher wishes to determine if the proportion of adults over the age of 65 in the city who plan to vote Republican in the next mayoral election has changed. Let p represent the proportion of the population of all adults over the age of 65 in the city who plan to vote Republican in the next mayoral election. In terms of p, the researcher should test which null and alternative hypotheses?

Example 2

• The square footage of the several thousand apartments in a new development is advertised to be 1250 square feet, on average. A tenant group thinks that the apartments are smaller than advertised. They hire an engineer to measure a sample of apartments to test their suspicions. Let m represent the true average area (in square feet) of these apartments. What are the appropriate null and alternative hypotheses?

Example 3

• Is the mean height for all adult American males between the ages of 18 and 21 now over 6 feet? Let m represent the population mean height of all adult American males between the ages of 18 and 21. What are the appropriate null and alternative hypotheses to answer this question?

Summary

• Null hypothesis. The null hypothesis, denoted by H0, is usually the hypothesis that sample observations result purely from chance.

– Statement that you try to find evidence against– Statement of no difference– Statement of no effect

Summary

• Alternative hypothesis. The alternative hypothesis, denoted by H1 or Ha, is the hypothesis that sample observations are influenced by some non-random cause.

– Statement that you try to find evidence for– Statement of difference– Statement of effect

Summary

• Solving a Hypothesis test problem involves 5 steps:– Formulate the null and alternative hypothesis– Choose a significance level– Compute a test statistic – Determine a p-value and compare it to the

significance level– Make conclusions

Example

• Weekly production of the Model A325 desk at the Fredonia Plant is normally distributed with a mean of 200 and a std dev of 16. The vice president of the company would like to investigate whether there has been a change in the weekly production of the Model A325 desk. What can he conclude at the 1% level of significance.

• Use n = 50, and sample mean = 203.5

Example

Step 1: Ho: μ = 200 vs Ha: μ ≠ 200

Step 2: α = .01

Example

Step 3: test statistic is computed from the formula:

z = ((sample mean) – (hypothesized mean))/ standard error

• We get: (203.5 – 200)/(16/sqrt(50)) = 1.55

Example

• Determine P-value: .12

• P-value > significance level

• i. e. .12 > .01

Conclusion: • Do not reject Ho. No evidence found to suspect

that the weekly mean production of A325 desk has changed from 200.