# Suppose a company claims that its market share is less than 16 percent, on average. Several of coworkers do not believe this, so a director decides to do a hypothesis test, at a 1% significance level, to persuade them. He conducts 21 surveys, collects the proper data, and works through the testing procedure:H0: μ≥16; Ha: μ<16x¯=15.8σ=1.8α=0.01 (significance level)The test statistic isz0=x¯−μ0σn√=15.8−161.821√=−0.51The critical value is −z0.01=−2.33.Conclude whether to reject or not reject H0, and interpret the results. Select the correct answer below: Reject H0. At the 1% significance level, the test results are not statistically significant and at best, provide weak evidence against the null hypothesis. Reject H0. At the 1% significance level, the data provide sufficient evidence to conclude that the mean market share is less than 16 percent. Do not reject H0. At the 1% significance level, the test results are not statistically significant and at best, provide weak evidence against the null hypothesis. Do not reject H0. At the 1% significance level, the data provide sufficient evidence to conclude that the mean market share is less than 16 percent.

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Suppose a company claims that its market share is less than 16 percent, on average. Several of coworkers do not believe this, so a director decides to do a hypothesis test, at a 1% significance level, to persuade them. He conducts 21 surveys, collects the proper data, and works through the testing procedure:

• H0: μ≥16; Ha: μ<16
• x¯=15.8
• σ=1.8
• α=0.01 (significance level)
• The test statistic is
z0=x¯−μ0σn√=15.8−161.821√=−0.51
• The critical value is −z0.01=−2.33.

Conclude whether to reject or not reject H0, and interpret the results.

Reject H0. At the 1% significance level, the test results are not statistically significant and at best, provide weak evidence against the null hypothesis.

Reject H0. At the 1% significance level, the data provide sufficient evidence to conclude that the mean market share is less than 16 percent.

Do not reject H0. At the 1% significance level, the test results are not statistically significant and at best, provide weak evidence against the null hypothesis.

Do not reject H0. At the 1% significance level, the data provide sufficient evidence to conclude that the mean market share is less than 16 percent.

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Step 1

The level of significance is 0.01.

The test statistic is z = -0.51.

The critical value is -2.33.

This test is a left tailed test.

Decision rule:

For positive test statistic, if , reject the null hypothesis, otherwise fail to reject the null hypothesis.

For negative test statistic, if , reject the null hypothesis, otherwise fail to reject the null hypothesis.

Step 2

Conclusion:

Here z = -0.51, critical value zα = -2.33.

Therefore, the test statistic is not less than the critical value.

That is, -0.51 > -2.33.

Using the decision rule, fail to reject the null hypothesis at 1% level of significance.

Hence, it can be concluded that there is no sufficient evidence to conclude that the market share is less than 16% at 1% level of significance.

Step 3

Therefore, from the given option, the third interpretation is correct.

Now, the 1st interpretation contradicts the 3rd interpretation. This means that the 1st interpretation must be incorrect when the 3rd interpretation is correct.

In similar way, 2nd interpretation contradicts the 3rd interpretation. This means that the 2nd interpretation must be incorrect when the 3rd interpretation is correct....

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### Hypothesis Testing 