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 other than 0.65 ? Use 4
decimal places for .
YES
NO
Note: Round any -score to 2 decimal places before any calculation.

Ask by Goodwin Hill. 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 other than 0.65 ? Use 4
decimal places for .
YES
NO
Note: Round any -score to 2 decimal places before any calculation.

Ask by Goodwin Hill. in the United States

Mar 21,2025

UpStudy AI · Accepted Answer

**Step 1. State the Hypotheses**

Let $$ p $$ be the true proportion of teen Facebook users who include photos of themselves. We have  
$$
H_0: p = 0.65 \quad \text{(null hypothesis)}
$$
$$
H_a: p \neq 0.65 \quad \text{(alternative hypothesis)}
$$

**Step 2. Calculate the Sample Proportion**

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 given by  
$$
\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.65 \cdot 0.35}{600}}.
$$
First, compute the product in the numerator:  
$$
0.65 \times 0.35 = 0.2275.
$$
Then, divide by $$600$$:  
$$
\frac{0.2275}{600} \approx 0.00037917.
$$
Taking the square root gives:  
$$
\sigma_{\hat{p}} \approx 0.0195 \quad \text{(to 4 decimal places)}.
$$

**Step 4. Calculate the Test Statistic (z-score)**

The z-score is computed by  
$$
z = \frac{\hat{p} - p}{\sigma_{\hat{p}}} = \frac{0.6583 - 0.65}{0.0195} \approx \frac{0.0083}{0.0195} \approx 0.43.
$$
Per instruction, round the z-score to 2 decimal places: $$ z \approx 0.43 $$.

**Step 5. Determine the Critical Value**

Since the test is two-tailed with a significance level of $$\alpha = 0.008$$, each tail has an area of $$\alpha/2 = 0.004$$. The critical z-values are:  
$$
z_{critical} \approx \pm 2.65.
$$

**Step 6. Make the Decision**

We fail to reject the null hypothesis if  
$$
-2.65 < z < 2.65.
$$
Since $$ 0.43 $$ lies within this range, we do not have sufficient evidence to reject $$ H_0 $$.

**Conclusion**

At a significance level of $$\alpha = 0.008$$, the sample does not show strong evidence that $$ p $$ is other than 0.65.

NO
At a significance level of 0.008, the sample does not provide strong evidence that the proportion $$ p $$ is different from 0.65. Therefore, the answer is **NO**.

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