Question

A stats quiz is known to have a mean score of 68 along with the standard deviation of 13. A class of 19 students takes quiz with a mean score of 65. (Two tail and α = 0.1)

2.1 For the above problem the Null and Alternate hypothesis is:

2.2 The calculated Z value is:

2.3 The critical Z value is:

2.4 The result is: (Fail to reject or Reject Null hypothesis)

2.5 The conclusion is:

Answer 1

One-Sample Z test

The sample mean is Xˉ=65, the population standard deviation is σ=13, and the sample size is n=19. (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: μ =68 Ha: μ ≠68 This corresponds to a Two-tailed test, for which a z-test for one mean, with known population standard deviation will be used. (2) Test Statistics The z-statistic is computed as follows: (3a) Critical Value Based on the information provided, the significance level is α=0.1, therefore the critical value for this Two-tailed test is Zc​=1.6449. This can be found by either using excel or the Z distribution table. (3b) Rejection Region The rejection region for this Two-tailed test is |Z|>1.6449 i.e. Z>1.6449 or Z<-1.6449 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|>1.0059)=0.3145 (4) The Decision about the null hypothesis (a) Using traditional method Since it is observed that |Z|=1.0059 < Zc​=1.6449, it is then concluded that the null hypothesis is Not rejected. (b) Using p-value method Using the P-value approach: The p-value is p=0.3145, and since p=0.3145>0.1, 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 Not enough evidence to claim that the population mean μ  is different than 68, at the 0.1 significance level.

Let me know in the comments if anything is not clear. I will reply ASAP! Please do upvote if satisfied!