Question

A test is made of H0 : μ = 55 versus H1 : μ > 55. A sample of size n=62 is drawn, and x=62. The population standard deviation is σ=27. Compute the value of the test statistic z and determine if H0 is rejected at the level.

Answer 1

Test statistic = z = 0.2593

p-value = 0.397702

Significance level = 0.05

0.397702 > 0.05

p-value > significance level

So, we fail to reject the null hypothesis.

H0 is not rejected at this level.

Step-by-step explanation

For hypothesis testing, the first thing to define is the null and alternative hypothesis.

The null hypothesis plays the devil's advocate and usually takes the form of the opposite of the theory to be tested. It usually contains the signs =, ≤ and ≥ depending on the directions of the test.

While, the alternative hypothesis usually confirms the the theory being tested by the experimental setup. It usually contains the signs ≠, < and > depending on the directions of the test.

For this question,

Null hypothesis is that

H0 : μ = 55

Alternative hypothesis is that

H1 : μ > 55

The test statistic is calculated as

z = (x - μ)/σₓ

x = Sample mean = 62

μ = Mean that we are comparing against = 55

σₓ = population standard deviation = 27

z = (x - μ)/σₓ

z = (62 - 55)/27

z = 0.2593

To now know the conclusion of the test, we need to comoute the p-value of the data

p-value of this value of z (0.2593), for a one-tailed test = 0.397702

The interpretation of p-values is that

When the (p-value > significance level), we fail to reject the null hypothesis and when the (p-value < significance level), we reject the null hypothesis and accept the alternative hypothesis.

For this test, when significance level is not given, we use 0.05

p-value = 0.397702

Significance level = 0.05

0.397702 > 0.05

p-value > significance level

So, we fail to reject the null hypothesis.

Hope this Helps!!!