Question

A simple random sample of 40 accounts is taken from an account receivables portfolio of ABC Ltd and the average account balance is $750. The population standard deviation σ is known to be $80. (6 points)

1. Test the hypothesis that the population mean account balance is greater than $680 using the p-value approach and a 0.05 level of significance.
2. Test the hypothesis that the population mean account balance is less than $820 using the critical value approach and a 0.05 level of significance.
3. Test the hypothesis that the population mean account balance is different from $800. using the p-value approach and a 0.05 level of significance

Answer 1

a

One-Sample Z test

The sample mean is Xˉ=750, the population standard deviation is σ=80, and the sample size is n=40. (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: μ =680 Ha: μ >680 This corresponds to a Right-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=0.05, therefore the critical value for this Right-tailed test is Zc​=1.6449. This can be found by either using excel or the Z distribution table. (2b) Rejection Region The rejection region for this Right-tailed test is Z>1.6449 (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>5.534)=0 (5) The Decision about the null hypothesis Using p-value method Using the P-value approach: The p-value is p=0, and since p=0≤0.05, 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 greater than 680, at the 0.05 significance level.

b)

One-Sample Z test

The sample mean is Xˉ=750, the population standard deviation is σ=80, and the sample size is n=40. (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: μ =820 Ha: μ <820 This corresponds to a Left-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=0.05, therefore the critical value for this Left-tailed test is Zc​=-1.6449. This can be found by either using excel or the Z distribution table. (2b) Rejection Region The rejection region for this Left-tailed test is Z<-1.6449 (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<-5.534)=0 (5) The Decision about the null hypothesis Using traditional method Since it is observed that Z=-5.534 < Zc​=-1.6449, it is then 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 less than 820, at the 0.05 significance level.

c)

One-Sample Z test

The sample mean is Xˉ=750, the population standard deviation is σ=80, and the sample size is n=40. (1) Null and Alternative Hypotheses The following null and alternative hypotheses need to be tested: Ho: μ =800 Ha: μ ≠800 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=0.05, therefore the critical value for this Two-tailed test is Zc​=1.96. 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|>1.96 i.e. Z>1.96 or Z<-1.96 (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|>3.9528)=0.0001 (5) The Decision about the null hypothesis (a) Using traditional method Since it is observed that |Z|=3.9528 > Zc​=1.96, 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=0.0001, and since p=0.0001≤0.05, 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 800, at the 0.05 significance level.