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Hypothesis Testing of Mean

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Historically the average age of European soccer players is reported as 26 years with a standard deviation of 4 years. A random sample of 81 European professional soccer players has an average age of 27 years. We would like to decide if there is enough evidence to establish that average age of European soccer players has increased significantly. What is the decision at ?=.05 and 0.01? Indicate which test you are performing; show the hypotheses, the test statistic and the critical values and mention whether one-tailed or two-tailed.

https://brainmass.com/statistics/hypothesis-testing/hypothesis-testing-mean-462803

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Since the population standard deviation is known, we can use z test for mean.
We are testing whether the average age of European soccer players has increased significantly. Hence the test is one-tailed and is an upper tailed test.
The null hypothesis tested is
H0: Average age of European soccer players ≤ 26 years. (µ ≤ 26)
The alternative hypothesis is
H1: Average age of European soccer players > 26 years. (µ > 26)
The test statistic used is , where =27, n = 81,  = 4
Therefore, = 2.25
When significance level, α = 0.05
Rejection criteria: Reject the null hypothesis, if the calculated value of test statistic is greater than the critical value at the 0.05 significance level.
Upper critical value = 1.644853627
Conclusion: Reject the null hypothesis, since the calculated value of test statistic is greater than the critical value. The sample provides enough evidence to support the claim that the average age of European soccer players has increased significantly.
Details
Z Test of Hypothesis for the Mean

Data
Null Hypothesis = 26
Level of Significance 0.05
Population Standard Deviation 4
Sample Size 81
Sample Mean 27

Intermediate Calculations
Standard Error of the Mean 0.444444444
Z Test Statistic 2.25

Upper-Tail Test
Upper Critical Value 1.644853627
p-Value 0.012224473
Reject the null hypothesis

When significance level, α = 0.01
Rejection criteria: Reject the null hypothesis, if the calculated value of test statistic is greater than the critical value at the 0.01 significance level.
Upper critical value = 2.326347874
Conclusion: Fails to reject the null hypothesis, since the calculated value of test statistic is less than the critical value. The sample does not provide enough evidence to support the claim that the average age of European soccer players has increased significantly.
Details
Z Test of Hypothesis for the Mean

Data
Null Hypothesis = 26
Level of Significance 0.01
Population Standard Deviation 4
Sample Size 81
Sample Mean 27

Intermediate Calculations
Standard Error of the Mean 0.444444444
Z Test Statistic 2.25

Upper-Tail Test
Upper Critical Value 2.326347874
p-Value 0.012224473
Do not reject the null hypothesis

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