Question

a recent publication claims that the average person spends 93 minutes per day on tiktok. a sociologist believes that adults in the baby boomer age range, spend less time on tiktok than the reported 93 minutes per day. a random sample of 18 adults in the baby boomer age found the mean time on tiktok to be 87.656 minutes with a standard deviation of 8.0256 minutes. assume that the times were normally distributed. test the sociologists claim using a 5% level of significance. give answer to at least 4 decimal places.
a. what are the correct hypotheses?
$h_0$: select an answer? $square$ minutes
$h_1$: select an answer? $square$ minutes
based on the hypotheses, find the following:
b. test statistic = $square$
c. p - value=$square$
d. the correct decision is to select an answer
e. the correct summary would be: select an answer that the population mean of adults in the baby boomer age range, spend less time on tiktok than the reported 93 minutes per day.

Explanation:

Step1: State the hypotheses

The null hypothesis $H_0$ is the statement of no - effect or no difference. The sociologist believes that baby - boomers spend less time on TikTok than 93 minutes. So, $H_0:\mu\geq93$ and $H_1:\mu < 93$.

Step2: Calculate the test statistic

The formula for the one - sample t - test statistic is $t=\frac{\bar{x}-\mu}{s/\sqrt{n}}$, where $\bar{x} = 87.656$ (sample mean), $\mu = 93$ (hypothesized population mean), $s = 8.0256$ (sample standard deviation), and $n = 18$ (sample size).

$$t=\frac{87.656 - 93}{8.0256/\sqrt{18}}$$

$$t=\frac{- 5.344}{8.0256/4.2426}$$

$$t=\frac{-5.344}{1.8917}$$

$$t\approx - 2.825$$

Step3: Calculate the p - value

Since this is a left - tailed test with degrees of freedom $df=n - 1=18 - 1 = 17$, we use the t - distribution to find the p - value. Using a t - distribution table or a calculator (e.g., in R: pt(-2.825,17)), the p - value is approximately $0.0069$.

Step4: Make a decision

The level of significance is $\alpha = 0.05$. Since the p - value ($0.0069$) is less than $\alpha=0.05$, we reject the null hypothesis.

Step5: Write the summary

There is sufficient evidence to support the claim that the population mean of adults in the baby - boomer age range spend less time on TikTok than the reported 93 minutes per day.

Answer:

a. $H_0:\mu\geq93$; $H_1:\mu < 93$
b. $-2.8250$
c. $0.0069$
d. Reject $H_0$
e. There is sufficient evidence