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 ...