A random sample of 100 observations from a quantitative population produced a sa

Question

A random sample of 100 observations from a quantitative population produced a sample mean of 30.8 and a sample standard deviation of 8.2. Use the p-value approach to determine whether the population mean is different from 32. Explain your conclusions. (Use a 0.05. Round your test statistic to two decimal places and your p-value to four decimal places.) 1-2. Null and alternative hypotheses: Ho: 32 versus Ha: 32 O Ho: -32 versus Ha: 32 H0: 32 versus Hai 32 Ho: 32 3. Test statistic: 4. p-value- O Ho is not rejected. There is insufficient evidence to indicate that the mean is different from 32. O Ho is rejected. There is sufficient evidence to indicate that the mean is different from 32. O Ho is not rejected. There is sufficient evidence to indicate that the mean is different from 32. O Ho is rejected. There is insufficient evidence to indicate that the mean is different from 32. ew

Explanation / Answer

Given :-

sample mean is X¯=30.8 and sample standard deviation is s=8.2, and the sample size is n = 100.

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: =32

Ha: 32

This corresponds to a two-tailed test, for which a z-test for one mean.

(2) Rejection Region

Based on the information provided, the significance level is =0.05, and the critical value for a two-tailed test is zc=1.96.

The rejection region for this two-tailed test is R={z:z>1.96}

(3) Test Statistics

The z-statistic is computed as follows:

z = [ (X¯0) / (s/n) ] = [ (30.832) / (8.2/100) ] = 1.46

(4) Decision about the null hypothesis

Since it is observed that z=1.46 zc=1.96, it is then concluded that the null hypothesis is not rejected.

P-value : The p-value is p=0.1434, and since p=0.1434 0.05, it is concluded that the null hypothesis is not rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is not rejected. Therefore, there is insufficient evidence to claim that the mean is different from 32, at the 0.05 significance level.

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