Please See Attached For Full Data On Young Children Using Ce

Please See Attached For Full Datado Young Children Use Cell Phones

Please see attached for full data. No sample sizes were reported. Suppose that the results were based on samples of 50 cell phone users in each group and that the sample standard deviation for cell phone users under 12 years of age was 51.7 calls per month and the sample standard deviation for cell phone users 13 to 17 years of age was 67.6 calls per month.

a. Using a 0.05 level of significance, is there evidence of a difference in the variances of cell phone usage between cell phone users under 12 years of age and cell phone users 13 to 17 years of age?

b. On the basis of the results in (a), which t test defined in Section 10.1 should you use to compare the means of the two groups of cell phone users? Discuss.

Paper For Above instruction

The utilization of cell phones among young children has become a significant topic of interest for researchers, especially given the increasing penetration of mobile technology into Daily life. A recent study aimed to investigate the differences in cell phone usage patterns between two age groups: children under 12 years and adolescents aged 13 to 17 years. Although specific sample sizes were not initially reported, an estimated sample size of 50 users per group was assumed for statistical analysis. The study provided means of 137 calls per month for children under 12 and 231 calls per month for teenagers, with standard deviations of 51.7 and 67.6, respectively.

Statistical Analysis of Variances (Part a)

The first step in understanding the data involves testing whether the variances of the two age groups' cell phone usage are significantly different. This is essential because many statistical tests assume equal variances—specifically, the Student’s t-test for independent samples. To examine this, an F-test for variances is employed. The null hypothesis states that the variances are equal, while the alternative hypothesis suggests they are not.

The test statistic for the F-test is calculated by dividing the larger variance by the smaller variance:

\[ F = \frac{S_2^2}{S_1^2} \]

where \( S_2^2 \) and \( S_1^2 \) are the sample variances of the two groups.

Calculating the variances:

\[ S_1^2 = (51.7)^2 = 2674.89 \]

\[ S_2^2 = (67.6)^2 = 4571.76 \]

Assuming the group with the greater variance is used as numerator:

\[ F = \frac{4571.76}{2674.89} \approx 1.71 \]

The critical value for the F-distribution at \( \alpha = 0.05 \), with degrees of freedom \( df_1 = 49 \) and \( df_2 = 49 \), can be obtained from F-tables or statistical software. The critical values are approximately 1.68 and 0.59 for the upper and lower bounds, respectively, for two-sided testing.

Since the computed F (1.71) exceeds the upper critical value (1.68), we reject the null hypothesis of equal variances. Therefore, we conclude there is statistically significant evidence at the 0.05 level to suggest that the variances of cell phone usage are different between children under 12 and adolescents aged 13–17.

Choosing the Appropriate t-test (Part b)

The next step involves selecting the appropriate t-test for comparing the means between the two groups, based on the results of the variance test. Traditional Student's t-test assumes equal variances, whereas Welch's t-test does not require this assumption and is more reliable when variances are unequal.

Given that the F-test indicates significant difference in variances, the appropriate test for comparing the means is Welch’s t-test. This test adjusts the degrees of freedom to account for variance inequality, providing a more accurate p-value.

Discussion

Employing Welch’s t-test ensures the validity of the statistical inference when variances are unequal—a situation confirmed by the preliminary F-test. In this context, the difference in variances suggests heterogeneity in usage variability between younger children and teenagers. Teenagers tend to have a broader range of cell phone call frequencies, indicated by the larger standard deviation, possibly reflecting more varied usage patterns influenced by social and academic factors.

In conclusion, the choice of Welch’s t-test is justified based on the initial variance comparison. It accounts for the unequal variances and provides a robust method to test for differences in mean cell phone usage between these distinct age groups, which has implications for understanding communication patterns among youth and for designing age-appropriate communication policies or interventions.

References

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