# Hypothesis testing problems

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Charnelle McDougald Week 9

10.30) A random sample of size n1 = 25, taken from a normal population with a standard deviation σ1 = 5.2, has a mean x1¯ = 81. A second random sample of size n2 = 36, taken from a different normal population with a standard deviation σ2 = 3.4, has a mean x2¯= 76. Test the hypothesis that µ1 = µ2 against the alternative µ1 ≠ µ2. Quote a P- value in your conclusion.

Answer

The test statistic used is

Where

Rejection criteria: Reject the null hypothesis, if the p-value is less than the significance level 0.05.

Details

t Test for Differences in Two Means

Data

Hypothesized Difference 0

Level of Significance 0.05

Population 1 Sample

Sample Size 25

Sample Mean 81

Sample Standard Deviation 5.2

Population 2 Sample

Sample Size 36

Sample Mean 76

Sample Standard Deviation 3.4

Intermediate Calculations

Population 1 Sample Degrees of Freedom 24

Population 2 Sample Degrees of Freedom 35

Total Degrees of Freedom 59

Pooled Variance 17.85694915

Difference in Sample Means 5

t Test Statistic 4.544883031

Two-Tail Test

Lower Critical Value -2.000995361

Upper Critical Value 2.000995361

p-Value 2.77668E-05

Reject the null hypothesis

Conclusion: Since the p value is less than the significance level, we reject the null hypothesis. The sample provides enough evidence to conclude that µ1 ≠ µ2.

10.32) Amstat News (December 2004) lists median salaries for associate professors of statistics at research institutions and at liberal arts and other institutions in the United States. Assume a sample of 200 associate professors from research institutions having an average salary of $70,750 per year with a standard deviation of $6000. Assume also a sample of 200 associate professors from other types of institutions having an average salary of $65,200 with a standard deviation of $5000. Test the hypothesis that the mean salary for associate professors in research institutions is $2000 higher than for those in other institutions. Use a 0.01 level of significance.

Answer

The null hypothesis tested is

H0: The difference in mean salary for associate professors in research institutions and those in other institutions is less than or equal to $2000 (µ1 - µ2 ≤ 2000)

The alternative hypothesis is

H1: The mean salary for associate professors in research institutions is $2000 higher than for those in other institutions (µ1 - µ2 > 2000)

The test Statistic used is

where

Rejection criteria: Reject the null hypothesis, if the calculated value of Z is greater than the critical value of Z at 0.01 significance level.

Details

Z Test for Differences in Two Means

Data

Hypothesized Difference 2000

Level of Significance 0.01

Population 1 Sample

Sample Size 200

Sample Mean 70750

Population Standard Deviation 6000

Population 2 Sample

Sample Size 200

Sample Mean 65200

Population Standard Deviation 5000

Intermediate Calculations

Difference in Sample Means 5550

Standard Error of the Difference in Means 552.2680509

Z-Test Statistic 6.428037969

Upper-Tail Test

Upper Critical Value 2.326347874

p-Value 6.46307E-11

Reject the null hypothesis

Conclusion: Reject the null hypothesis. The sample provide enough evidence to support the claim that the mean salary for associate professors in research institutions is $2000 higher than for those in other institutions.

10.36). A large automobile manufacturing company is trying to decide whether to purchase brand A or brand B tires for its new models. To help arrive at a decision, an experiment is conducted using 12 of each brand. The tires are run until they wear out. The results are:

Brand A: x1¯= 37,900 kilometers,

s1 = 5,100 kilometers

Brand B: x2¯ = 39,800 kilometers

s2 = 5,900 kilometers

Test the hypothesis that there is no difference in the average wear of 2 brands of tires. Assume the populations to be approximately normally distributed with equal variances. Use a P-value.

Answer

The null hypothesis tested is

H0: There is no significant difference in the average wear of 2 ...

#### Solution Summary

The solution provides step-by-step method for the calculation of test statistic. Formula for the calculation and Interpretations of the results are also included.