Question

in a survey conducted by a reputable marketing agency, 245 of 1000 adults 19 years of age or older confessed to bringing and using their cell phone every trip to the bathroom (confessions included texting and answering phone calls). complete parts (a) through (f) below

what is the source of variability in the random variable?

a. the sample size

b. the marketing agency

c. the question asked in the survey

d. the individuals selected to be in the study

(e) construct and interpret a 95% confidence interval for the population proportion of adults 19 years of age or older who bring their cell phone every trip to the bathroom. select the correct choice below and fill in any answer boxes within your choice.
(type integers or decimals rounded to three decimal places as needed. use ascending order.)

a. there is a % probability the proportion of adults 19 years of age or older who bring their cell phone every trip to the bathroom is between and

b. we are % confident the proportion of adults 19 years of age or older who bring their cell phone every trip to the bathroom is between and

Explanation:

Step1: Calculate the sample proportion ($\hat{p}$)

The sample proportion $\hat{p}=\frac{x}{n}$, where $x = 245$ (number of successes) and $n=1000$ (sample size). So, $\hat{p}=\frac{245}{1000}=0.245$.

Step2: Check the normality conditions

For a confidence interval of proportion, we need $n\hat{p}\geq5$ and $n(1 - \hat{p})\geq5$.
$n\hat{p}=1000\times0.245 = 245\geq5$ and $n(1-\hat{p})=1000\times(1 - 0.245)=1000\times0.755 = 755\geq5$.

Step3: Find the critical value ($z_{\alpha/2}$)

For a 95% confidence interval, $\alpha=1 - 0.95=0.05$, and $\alpha/2=0.025$.
From the standard normal distribution table, $z_{\alpha/2}=z_{0.025}=1.96$.

Step4: Calculate the margin of error ($E$)

The formula for the margin of error for a proportion is $E = z_{\alpha/2}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$.
Substitute $\hat{p}=0.245$, $n = 1000$, and $z_{\alpha/2}=1.96$ into the formula:

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Step5: Calculate the confidence interval

The confidence interval for the population proportion $p$ is $\hat{p}-ESubstitute $\hat{p}=0.245$ and $E = 0.027$:
$0.245-0.027=0.218$ and $0.245 + 0.027=0.272$.

Answer:

B. We are 95% confident the proportion of adults 19 years of age or older who bring their cell phone every trip to the bathroom is between 0.218 and 0.272.