A researcher believe that college students spend a different amount of time talking on the phone than they do instant messaging. She takes a random sample of 12 college students and records the number of hours per week each student spends on each activity and plans to test at the 5% level.
1. Based on the description of how the data were gathered, which test design is appropriate? (Paired Samples T Test or Independent Samples T Test)
2. What is the parameter of interest? Describe in words in the context of the problem.
3. For the test to be valid, what TWO assumptions must be made?
4. For the sample of 12 students, the mean number of hours of phone use per week was 27.9 and the standard deviation was 1.6. The mean number of hours of instant messaging per week was 25.3 and the standard deviation was 3.7. The sample mean difference was 2.7 and the standard deviation of the differences was 3.3. Calculate the test statistic. Show your work.
5. Is this a one-sided or two-sided test? (One or Two Sided)
6. What is the value of the degrees of freedom for this test?
7. Using your Appendix, give bounds for the p-value for this test.
8. Based on the p-value you calculated, what is your statistical decision? (Reject or Do Not Reject the Null).
9. Provide a conclusion in the context of the problem
10. Give the null and alternative hypotheses for this test
1. The appropriate test design for this scenario is the Paired Samples T Test because the researcher is comparing two sets of data from the same group of college students.
2. The parameter of interest is the mean difference in the number of hours spent talking on the phone and instant messaging per week among college students.
3. The two assumptions that must be made for the test to be valid are:
a) The differences in the paired observations are normally distributed.
b) The variances of the paired observations are approximately equal.
4. To calculate the test statistic, we can use the formula: t = (mean difference - hypothesized mean difference) / (standard deviation of the differences / square root of sample size). Plugging in the values, t = (2.7 - 0) / (3.3 / √12) = 2.7 / (3.3 / 3.464) = 2.7 / 0.953 = 2.828.
5. This is a two-sided test because we are testing if the mean difference is significantly different from zero, without specifying a particular direction.
6. The degrees of freedom for this test is (n - 1), where n is the sample size. In this case, the sample size is 12, so the degrees of freedom is 11.
7. Since the exact p-value cannot be calculated without the full data set, we will need to refer to an Appendix or a T-test table to find the bounds for the p-value. Typically, the table provides critical values for different levels of significance (e.g., 0.05, 0.01, etc.) and degrees of freedom.
8. Based on the calculated test statistic and the p-value obtained from the table, we can make our statistical decision. If the p-value is less than the significance level (5% in this case), we would reject the null hypothesis. If the p-value is greater than the significance level, we would fail to reject the null hypothesis.
9. In the context of the problem, based on the statistical decision, we can conclude that there is evidence to suggest a significant difference in the amount of time college students spend talking on the phone and instant messaging per week.
10. Null hypothesis (H0): The mean difference in the number of hours spent talking on the phone and instant messaging per week among college students is equal to zero.
Alternative hypothesis (Ha): The mean difference in the number of hours spent talking on the phone and instant messaging per week among college students is not equal to zero.
1. Based on the description of how the data were gathered, the appropriate test design would be the Paired Samples T Test. This is because the researcher collected data from the same group of students, taking measurements of phone use and instant messaging from each individual student.
2. The parameter of interest in this problem is the mean difference in the number of hours spent on phone use and instant messaging for college students.
3. The two assumptions that must be made for the test to be valid are:
a. The differences between phone use and instant messaging are normally distributed in the population.
b. The differences between phone use and instant messaging for each student are independent of each other.
4. To calculate the test statistic, we use the formula:
t = (mean of differences - hypothesized mean difference) / (standard deviation of the differences / sqrt(sample size))
Substituting the given values:
t = (2.7 - 0) / (3.3 / sqrt(12))
t = 2.7 / (3.3 / 3.464)
t ≈ 2.7 / 0.951
t ≈ 2.836
5. This is a two-sided test because we are interested in whether the mean difference is significantly different from zero, regardless of the direction.
6. The degrees of freedom for this test is given by (sample size - 1), so in this case, the degrees of freedom would be (12 - 1) = 11.
7. The exact bounds for the p-value can be found using a t-distribution table or statistical software. Since no Appendix is provided in the question, it's not possible to provide specific bounds for the p-value.
8. Based on the calculated p-value, if it is less than the significance level (5%), the statistical decision would be to Reject the Null hypothesis. If it is greater than or equal to the significance level, the decision would be to Not Reject the Null hypothesis.
9. The conclusion in the context of the problem would depend on the statistical decision. If the Null hypothesis is rejected, it would indicate that there is a significant difference in the amount of time college students spend on phone use and instant messaging. If the Null hypothesis is not rejected, it would indicate that there is not enough evidence to suggest a significant difference.
10. The null hypothesis (H0) would be that there is no difference in the mean number of hours spent on phone use and instant messaging for college students. The alternative hypothesis (Ha) would be that there is a difference in the mean number of hours spent on phone use and instant messaging for college students.