Question

a recent publication claims that the average person spends 95 minutes per day on tiktok. a sociologist believes that adults in the baby boomer age range, spend less time on tiktok than the reported 95 minutes per day. a random sample of 16 adults in the baby boomer age found the mean time on tiktok to be 90.131 minutes with a standard deviation of 9.3581 minutes. assume that the times were normally distributed. test the sociologists claim using a 1% level of significance. give answer to at least 4 decimal places.
a. what are the correct hypotheses?
$h_0$: select an answer?
$h_1$: select an answer?
based on the hypotheses, find the following:
b. test statistic =
c. p - value=
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 95 minutes per day.

Explanation:

Step1: Define null and alternative hypotheses

The null hypothesis $H_0$ is the status - quo, and the alternative hypothesis $H_1$ is what the sociologist claims. The sociologist claims that baby - boomers spend less time on TikTok than 95 minutes per day. So, $H_0:\mu\geq95$ and $H_1:\mu < 95$, where $\mu$ is the population mean time baby - boomers spend on TikTok per day.

Step2: Calculate the test statistic

The formula for the one - sample t - test statistic is $t=\frac{\bar{x}-\mu_0}{s/\sqrt{n}}$, where $\bar{x} = 90.131$ is the sample mean, $\mu_0 = 95$ is the hypothesized population mean, $s = 9.3581$ is the sample standard deviation, and $n = 16$ is the sample size.

$$t=\frac{90.131 - 95}{9.3581/\sqrt{16}}=\frac{- 4.869}{9.3581/4}=\frac{-4.869}{2.339525}\approx - 2.0812$$

Step3: Calculate the p - value

Since this is a left - tailed test with $n-1=16 - 1 = 15$ degrees of freedom, we use the t - distribution to find the p - value. Using a t - distribution table or a calculator (e.g., in R: pt(-2.0812,15)), the p - value is $P(t_{15}< - 2.0812)\approx0.0277$.

Step4: Make a decision

The level of significance is $\alpha = 0.01$. Since the p - value ($0.0277$) > $\alpha(0.01)$, we fail to reject the null hypothesis.

Step5: Summarize the result

There is not enough evidence at the 1% level of significance to support the claim that the population mean of adults in the baby - boomer age range spend less time on TikTok than the reported 95 minutes per day.

Answer:

a. $H_0:\mu\geq95$; $H_1:\mu < 95$
b. $-2.0812$
c. $0.0277$
d. Fail to reject $H_0$
e. There is not enough evidence