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Analyze a study examining whether the average weekly television watching hours of children aged 2 to 5 has increased from 27 hours in 2000. A sample of 100 households reported an average of 31 hours watched per week with a standard deviation of 9 hours. The assignment involves calculating the standard error, formulating hypotheses, computing the test statistic, interpreting the results, determining the p-value, making a decision at a 5% significance level, and providing a conclusion about the findings. Additionally, it addresses the importance of the probability of Type I error in the context of the study.

Paper For Above instruction

The research question in this study centers on determining whether there has been a significant increase in the amount of television children watch weekly from the year 2000 to the present period. The historical data indicates that, in 2000, children aged 2 to 5 watched an average of 27 hours per week. The recent data collected from 100 households suggests an increase to an average of 31 hours, with a standard deviation of 9 hours. The study aims to evaluate whether this observed difference is statistically significant, implying a potential change in viewing habits over the years.

The parameter of interest is the population mean number of hours children aged 2 to 5 watch television weekly in the current context. The null hypothesis posits that the mean weekly television viewing time has remained unchanged since 2000, indicating no increase; formally, H0: μ = 27 hours. The alternative hypothesis suggests that the mean has increased, expressed as H1: μ > 27 hours, representing a one-tailed test designed to detect any upward trend in television watching habits.

To test this hypothesis, the appropriate statistical procedure is a one-sample z-test for the mean, assuming the population standard deviation is unknown but the sample size is sufficiently large (n=100) for the Central Limit Theorem to apply. First, we compute the standard error (SE) of the mean, which estimates the variability that would be expected in the sample mean if repeated samples were drawn from the population. The standard error is calculated as:

SE = s / √n = 9 / √100 = 9 / 10 = 0.9 hours.

Next, the test statistic (z-value) is calculated using the formula:

z = (x̄ - μ₀) / SE

where x̄ = 31 hours (sample mean), μ₀ = 27 hours (historical mean under null hypothesis). Substituting, we get:

z = (31 - 27) / 0.9 ≈ 4.44.

This z-value measures the number of standard errors the sample mean is above the hypothesized population mean. Interpreted as a distance, a z of 4.44 indicates that the observed mean is 4.44 standard errors above the null hypothesized mean, signifying a significant deviation from the null value.

Under the null hypothesis, the distribution of the test statistic is approximately standard normal (a normal distribution with mean 0 and standard deviation 1). This assumption holds due to the large sample size (n=100), which ensures the Central Limit Theorem applies and validates the use of the normal approximation for the sampling distribution of the mean.

To determine the p-value, which indicates the probability of observing a test statistic as extreme or more extreme as the calculated z-value under the null, we consult the standard normal distribution. For z = 4.44, the p-value is extremely small, effectively approaching zero. Using statistical software like R with the pnorm() function, the p-value for a one-sided test is:

p_value = 1 - pnorm(4.44)

p_value ≈ 1.46e-05

This p-value is much less than the significance level of 0.05, indicating strong evidence against the null hypothesis.

At a 5% significance level, the decision is to reject the null hypothesis. The data provides sufficient evidence to conclude that the average weekly television viewing hours for children aged 2 to 5 have increased since 2000.

In summary, based on the statistical analysis, there is compelling evidence that children in this age group are now watching significantly more television on a weekly basis than they did two decades ago. This finding has important implications for parents, educators, and policymakers concerned with early childhood development and media exposure.

Additionally, the probability of incorrectly concluding that children watch more television when they do not (Type I error) is denoted by alpha (α). In this context, setting α at 0.05 corresponds to accepting a 5% risk of falsely rejecting the null hypothesis, which is standard practice in hypothesis testing and reflects the level of significance used for decision-making in this study.

References

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