Question

you recently sent out a survey to determine if the percentage of adults who use social media has changed from 53%, which was the percentage of adults who used social media five years ago. of the 2682 people who responded to survey, 1715 stated that they currently use social media.
a) use the data from this survey to construct a 98% confidence interval estimate of the proportion of adults who use social media. record the result below in the form of (#, #). the parenthesis and comma must be included. round your final answer to four decimal places.
b) can you conclude that the percentage of adults who use social media has changed? explain.
yes, because the proportion of adults who used social media five years ago is inside of the confidence interval.
yes, because the proportion of adults who used social media five years ago is not inside the confidence interval.
no, because the proportion of adults who used social media five years ago is inside the confidence interval.
no, because the proportion of adults who used social media five years ago is not inside the confidence interval.
c) if it has changed, has it increased or decreased?
the percentage increased.
the percentage decreased.
we cannot determine that the percentage has changed.

Explanation:

Step1: Calculate sample proportion $\hat{p}$

$\hat{p}=\frac{x}{n}$, where $x = 1715$ (number of successes) and $n=2682$ (sample size). So, $\hat{p}=\frac{1715}{2682}\approx0.6395$.

Step2: Find $z$-value for 98% confidence interval

The confidence level is 98%, so the significance level $\alpha=1 - 0.98 = 0.02$. Then $\alpha/2=0.01$. The $z$-value $z_{\alpha/2}=z_{0.01}\approx 2.33$.

Step3: 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 $\hat{p}=0.6395$, $n = 2682$ and $z_{\alpha/2}=2.33$.
$E=2.33\sqrt{\frac{0.6395\times(1 - 0.6395)}{2682}}\approx2.33\sqrt{\frac{0.6395\times0.3605}{2682}}\approx2.33\sqrt{\frac{0.2305}{2682}}\approx2.33\times0.0093\approx0.0216$.

Step4: Calculate the confidence - interval

The confidence interval is $\hat{p}-E

Step5: Answer part b

The proportion of adults who used social media five years ago is $p_0 = 0.53$. Since $0.53$ is not inside the confidence interval $(0.6179,0.6611)$, we can conclude that the proportion has changed. The answer is: Yes, because the proportion of adults who used social media five years ago is not inside the confidence interval.

Step6: Answer part c

Since the entire confidence interval $(0.6179,0.6611)$ is greater than $0.53$, the percentage has increased.

Answer:

a. $(0.6179,0.6611)$
b. Yes, because the proportion of adults who used social media five years ago is not inside the confidence interval.
c. The percentage increased.