Consider the following problem that tests a hypothesis:
A random sample of 600 teens with Facebook profiles found that 395 included photos of themselves. Let
be the proportion of teen Facebook users who included photos of themselves a profile pictures.
At significance level , is the sample showing strong evidence that is more than 0.58 ? Use 4
decimal places for
NO
YES
Note: Round any -score to 2 decimal places before any calculation.

Ask by Reeves Gibson. in the United States

Mar 21,2025

Question

Consider the following problem that tests a hypothesis:
A random sample of 600 teens with Facebook profiles found that 395 included photos of themselves. Let
be the proportion of teen Facebook users who included photos of themselves a profile pictures.
At significance level , is the sample showing strong evidence that is more than 0.58 ? Use 4
decimal places for
NO
YES
Note: Round any -score to 2 decimal places before any calculation.

Ask by Reeves Gibson. in the United States

Mar 21,2025

UpStudy AI · Accepted Answer

**Step 1. State the Hypotheses**  
We are testing whether the true proportion $$ p $$ exceeds 0.58. Thus, our hypotheses are:  
$$
H_0: p = 0.58 \quad \text{vs.} \quad H_a: p > 0.58
$$

**Step 2. Calculate the Sample Proportion**  
Given that 395 out of 600 teens had photos in their profiles, the sample proportion is  
$$
\hat{p} = \frac{395}{600} \approx 0.6583
$$

**Step 3. Compute the Standard Error**  
The standard error under the null hypothesis is computed by:  
$$
\sigma_{\hat{p}} = \sqrt{\frac{p_0(1-p_0)}{n}} = \sqrt{\frac{0.58 \times 0.42}{600}}
$$  
First, compute the product in the numerator:  
$$
0.58 \times 0.42 = 0.2436
$$  
Then,  
$$
\sigma_{\hat{p}} = \sqrt{\frac{0.2436}{600}} = \sqrt{0.0004060} \approx 0.0202
$$  
(using 4 decimal places for $$ \sigma_{\hat{p}} $$).

**Step 4. Compute the Test Statistic**  
The $$ z $$-score is given by:  
$$
z = \frac{\hat{p} - p_0}{\sigma_{\hat{p}}} = \frac{0.6583 - 0.58}{0.0202} = \frac{0.0783}{0.0202} \approx 3.88
$$  
(rounded to 2 decimal places).

**Step 5. Determine the p-value**  
Since the test is one-tailed (right-tailed), the p-value is:  
$$
\text{p-value} = P(Z > 3.88)
$$  
Looking up the value or using a calculator, we find that the p-value is very small (approximately 0.00005), which is much less than the significance level $$ \alpha = 0.005 $$.

**Step 6. Conclusion**  
Since the p-value is less than $$ \alpha $$, we reject the null hypothesis. There is strong evidence at the $$ \alpha = 0.005 $$ significance level that  
$$
p > 0.58.
$$

Yes
Yes, there is strong evidence that the proportion $$ p $$ is more than 0.58.

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