Consider the following hypotheses: H0: 79.0 HA: > 79.0 A sample of 45 observatio

Question

Consider the following hypotheses: H0: 79.0 HA: > 79.0 A sample of 45 observations yields a sample mean of 80.0. Assume that the sample is drawn from a normal population with a known population standard deviation of 5.1. Use Table 1. a. Calculate the p-value. (Round "z" value to 2 decimal places.) p-value b. What is the conclusion if = 0.01? Reject H0 Do not reject H0 c. Calculate the p-value if the above sample mean was based on a sample of 105 observations. (Round "z" value to 2 decimal places.) p-value d. What is the conclusion if = 0.01? Do not reject H0 Reject H0

Explanation / Answer

Solution:-

A sample of 45 observations yields a sample mean of 80.0. Assume that the sample is drawn from a normal population with a known population standard deviation of 5.1.

a. Calculate the p-value. (Round "z" value to 2 decimal places.) p-value.

The solution to this problem takes four steps: (1) state the hypotheses, (2) formulate an analysis plan, (3) analyze sample data, and (4) interpret results. We work through those steps below:

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: H0: 79.0
Alternative hypothesis: HA: > 79.0

Note that these hypotheses constitute a one-tailed test. The null hypothesis will be rejected if the sample mean is too large.

Formulate an analysis plan. For this analysis, the significance level is 0.01.

Analyze sample data. Using sample data, we compute the standard error (SE), degrees of freedom (DF), and the z statistic test..

SE = s / sqrt(n) = 5.1 / sqrt(45) = 0.76026
DF = n - 1 = 45 - 1 = 44

Z = (X - ) /
Z = (80 - 79) / 0.76026
Z = 1.32

where s is the standard deviation of the sample, x is the sample mean, is the hypothesized population mean, and n is the sample size.

Here is the logic of the analysis: Given the alternative hypothesis ( > 79), we want to know whether the observed sample mean is large enough to cause us to reject the null hypothesis.

Calculator to find P(z < 1.32)

The P-Value is 0.093418.
The result is not significant at p < 0.01.

b. What is the conclusion if = 0.01? Reject H0 Do not reject H0

Interpret results. Since the P-value is greater than the significance level (0.01), we cannot reject the null hypothesis.

c. Calculate the p-value if the above sample mean was based on a sample of 105 observations. (Round "z" value to 2 decimal places.) p-value.

SE = s / sqrt(n) = 5.1 / sqrt(105) = 0.497709
DF = n - 1 = 105 - 1 = 104

Z = (X - ) / SE
Z = (80 - 79) / 0.497709
Z = 2.01

The P-Value is 0.022216.
The result is not significant at p < 0.01.

d. What is the conclusion if = 0.01? Do not reject H0 Reject H0

Interpret results. Since the P-value is greater than the significance level (0.01), we cannot reject the null hypothesis.

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