Conduct a full hypothesis test and determine if the given claim is supported or not supported at the 0.05 significance level. Round all amounts and standard scores to two decimal places.
-In one city, the average amount of time that tenth-graders spend watching television each week is 22.4 hours. the principal of Birchwood High School believes that at his school, tenth-graders watch less television. For a
Sample of 32 tenth-graders from Birchwood High School, the mean amount of time spent watching television per week was 20.2 hours. Test whether the claim that for all tenth-graders at Birchwood High School, the mean
Amount of time spent watching television per week is less than the city average of 22.4 hours is supported or not supported. Assume that $\sigma = 7.2$
Hours.
A)not supported
B)supported

Question

Quizplus · Accepted Answer

The hypothesis test involves comparing the sample mean to the population mean to see if there is significant evidence that Birchwood High School tenth-graders watch less television than the city average. The null hypothesis ($H_0$) is that the mean amount of time Birchwood High School tenth-graders spend watching television is equal to or more than the city average ($\mu \geq 22.4$ hours), and the alternative hypothesis ($H_a$) is that the mean amount of time is less than the city average ($\mu < 22.4$ hours). Using the given data ($\bar{x} = 20.2$ hours, $n = 32$, $\sigma = 7.2$ hours), we calculate the z-score to determine if the difference is statistically significant at the $\alpha = 0.05$ level.The z-score is calculated as follows:$$ z = \frac{\bar{x} - \mu}{\sigma/\sqrt{n}} = \frac{20.2 - 22.4}{7.2/\sqrt{32}} \approx -2.19 $$Looking up the z-score in a standard normal distribution table or using a calculator, we find that the p-value associated with a z-score of -2.19 is less than 0.05, indicating that the result is statistically significant. Therefore, we reject the null hypothesis in favor of the alternative hypothesis, supporting the claim that Birchwood High School tenth-graders watch less television than the city average.

Conduct a Full Hypothesis Test and Determine If the Given σ=7.2\sigma = 7.2σ=7.2

Conduct a Full Hypothesis Test and Determine If the Given $\sigma = 7.2$