Question

5050 people are randomly selected and the accuracy of their wristwatches is checked, with positive errors representing watches that are ahead of the correct time and negative errors representing watches that are behind the correct time. The 5050 values have a mean of 111111sec and a population standard deviation of 233233sec. Use a 0.010.01 significance level to test the claim that the population of all watches has a mean of 00sec.

The test statistic is

The P-Value is

The final conclustion is

A. There is sufficient evidence to warrant rejection of the claim that the mean is equal to 0
B. There is not sufficient evidence to warrant rejection of the claim that the mean is equal to 0

Answer 1

One-Sample Z test

The sample mean is Xˉ=111, the population standard deviation is σ=233, and the sample size is n=50. (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: μ =0 Ha: μ ≠0 This corresponds to a Two-tailed test, for which a z-test for one mean, with known population standard deviation will be used. (2a) Critical Value Based on the information provided, the significance level is α=0.01, therefore the critical value for this Two-tailed test is Zc​=2.5758. This can be found by either using excel or the Z distribution table. (2b) Rejection Region The rejection region for this Two-tailed test is |Z|>2.5758 i.e. Z>2.5758 or Z<-2.5758 (3) Test Statistics The z-statistic is computed as follows: (4) The p-value The p-value is the probability of obtaining sample results as extreme or more extreme than the sample results obtained, under the assumption that the null hypothesis is true. In this case, the p-value is p =P(|Z|>3.3686)=0.0008 (5) A. There is sufficient evidence to warrant rejection of the claim that the mean is equal to 0 The Decision about the null hypothesis (a) Using traditional method Since it is observed that |Z|=3.3686 > Zc​=2.5758, it is then concluded that the null hypothesis is rejected. (b) Using p-value method Using the P-value approach: The p-value is p=0.0008, and since p=0.0008≤0.01, it is concluded that the null hypothesis is rejected. (6) Conclusion It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that the population mean μ  is different than 0, at the 0.01 significance level.

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