Please show all steps/calculations/equations in written format and a final state

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Please show all steps/calculations/equations in written format and a final statement for answer. Thanks

A study was conducted to determine if the salaries of the professors from two neighbouring universities were equal. A sample of 20 professors from each university was randomly selected. The mean from the first university was $109, 100 with a population standard deviation of $2300. The mean from the second university was $110, 500 with a population standard deviation of $2100. Assume that the distribution of professor salaries, at both universities, are approximately normally distributed. Level of significance = 0.05. a. Using the critical value approach test the claim that the salaries from both universities are equal. Include all key steps of the test. b. Use the four-step P-value approach and test the claim that the salaries from both universities are equal. c. Are the conditions met for using this test?

Explanation / Answer

a. Step1: Making assumptions and meeting test requirements

Model: Independent random samples; Level of measurement is interval ratio; Sampling distribution is normal.

Step2: Hypotheses: The null and alternative hy[potheses are as follows:

H0: mu1=mu2 (mean salaries for both universities are equal)

H1:mu1=/=mu2 (mean salaries for both universities are not equal)

Step3: Selecting sampling distribution and establishing the critical region. For known population standard deviations, and normally distributed population, use Z distribution to find areas under sampling distribution and establish the critical region. Alpha is set at 0.05.

Sampling distribution: Z; alpha=0.05; Z(critical)=+-1.96 (two-tailed test)

Step4. Compute test statistic.

Z=(X1bar-X2bar)/sqrt[sigma1^2/N1+sigma2^2/N2]

=(109100-110500)/sqrt[2300^2/10+2100^2/20]

=-1.62

Step5. Making a decision: If observed test statistic falls in critical region (observed Z>=critical Z), reject null hypothesis. Here, observed test statistic (-1.62) does not fall in critical region (-1.96), thus fail to reject null hypothesis.

Step6. Conclusion: There is insufficient sample evidence to reject the warrant of the claim that salaries from both universities are equal.

b. The p value approcah is as follows:

Step1: Hypotheses: The null and alternative hy[potheses are as follows:

H0: mu1=mu2 (mean salaries for both universities are equal)

H1:mu1=/=mu2 (mean salaries for both universities are not equal)

Step2: Selecting sampling distribution. For known population standard deviations, and normally distributed population, use Z distribution to find test statistic. Alpha is set at 0.05.

Sampling distribution: Z; alpha=0.05

Step3. Compute test statistic.

Z=(X1bar-X2bar)/sqrt[sigma1^2/N1+sigma2^2/N2]

=(109100-110500)/sqrt[2300^2/10+2100^2/20]

=-1.62

P value: 0.1052

Step4. Conclusion: Per rejection rule reject null hypothesis if p value is less than alpha=0.05. Here, p value is not less than 0.05, therefore, fail to reject null hypothesis. There is insufficient sample evidence to reject the warrant of the claim that salaries from both universities are equal.

c.

Model: Independent random samples; Level of measurement (salary) is interval ratio; Sampling distribution is normal (distribution of professor's salaries are approximately normal).

Therefore, the conditions are met for using the test.

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