Answered

IDNLearn.com is the perfect place to get detailed and accurate answers to your questions. Get accurate and detailed answers to your questions from our knowledgeable and dedicated community members.

The sponsors of television shows targeted at the market of [tex]$5$[/tex]- to [tex]$8$[/tex]-year-olds want to test the hypothesis that children watch television at most 20 hours per week. The population of viewing hours per week is known to be normally distributed with a standard deviation of 6 hours. A market research firm conducted a random sample of 30 children in this age group. At a 0.10 level of significance, use Excel to test the sponsors' hypothesis. The resulting data follows:

\begin{tabular}{|l|l|}
\hline Sample Size & 30 \\
\hline Sample Mean & 20.2667 \\
\hline Population Std. Dev. & 6 \\
\hline Hypothesized Mean & 20 \\
\hline Alpha & 0.1 \\
\hline
\end{tabular}

1. What is the Alternative Hypothesis [tex]\( H_a: \mu \neq 20 \)[/tex]

2. What is the Standard Error?

3. What is the Test Statistic (Z-Test value)?

4. What is the P-value?

5. What is the Conclusion (Reject or Accept [tex]\( H_0 \)[/tex])?

---

To help clarify:

1. [tex]\( H_a \)[/tex]: The alternative hypothesis is that the true mean viewing hours per week is not equal to 20.
2. Standard Error: Calculate the standard error of the sample mean.
3. Test Statistic: Compute the Z-Test value.
4. P-value: Determine the P-value corresponding to the test statistic.
5. Conclusion: Based on the P-value and the significance level, determine whether to reject or accept the null hypothesis [tex]\( H_0 \)[/tex].

Sagot :

GinnyAnsweridn
Let's proceed with clarity and detail through each step of the hypothesis test.

### Step 1: Stating the Hypotheses
The sponsors want to test the hypothesis that children watch television at most [tex]\(20\)[/tex] hours per week.

- The null hypothesis ([tex]\(H_0\)[/tex]): [tex]\(\mu \leq 20\)[/tex]
- The alternative hypothesis ([tex]\(H_a\)[/tex]): [tex]\(\mu > 20\)[/tex]

### Step 2: Calculating the Standard Error
The standard error (SE) is calculated using the formula:

[tex]\[ \text{SE} = \frac{\sigma}{\sqrt{n}} \][/tex]

Where:
- [tex]\(\sigma\)[/tex] is the population standard deviation ([tex]\(6\)[/tex] hours)
- [tex]\(n\)[/tex] is the sample size ([tex]\(30\)[/tex])

Plugging in the values:

[tex]\[ \text{SE} = \frac{6}{\sqrt{30}} \approx 1.0954 \][/tex]

### Step 3: Calculating the Z-Test Statistic
The z-test statistic is calculated using the formula:

[tex]\[ Z = \frac{\bar{x} - \mu_0}{\text{SE}} \][/tex]

Where:
- [tex]\(\bar{x}\)[/tex] is the sample mean ([tex]\(20.2667\)[/tex])
- [tex]\(\mu_0\)[/tex] is the hypothesized mean ([tex]\(20\)[/tex])
- [tex]\(\text{SE}\)[/tex] is the standard error ([tex]\(1.0954\)[/tex])

Plugging in the values:

[tex]\[ Z = \frac{20.2667 - 20}{1.0954} \approx 0.2435 \][/tex]

### Step 4: Calculating the P-value
Given the Z-test value, the next step is to find the corresponding p-value. In this case, because this is a one-tailed test where we're testing if the mean is greater than the hypothesized mean, we take:

[tex]\[ \text{P-value} = 1 - \text{CDF}(Z) \][/tex]

For [tex]\( Z = 0.2435 \)[/tex]:

[tex]\[ \text{P-value} \approx 0.4038 \][/tex]

### Step 5: Making the Decision
We compare the p-value with the significance level ([tex]\(\alpha = 0.1\)[/tex]):

- If [tex]\( \text{P-value} \leq \alpha \)[/tex], we reject [tex]\( H_0 \)[/tex].
- If [tex]\( \text{P-value} > \alpha \)[/tex], we fail to reject [tex]\( H_0 \)[/tex].

Here,

[tex]\[ \text{P-value} = 0.4038 \][/tex]
[tex]\[ \alpha = 0.1 \][/tex]

Since [tex]\(0.4038 > 0.1\)[/tex], we fail to reject the null hypothesis [tex]\( H_0 \)[/tex]. This means there is insufficient evidence to conclude that children watch more than [tex]\( 20 \)[/tex] hours of television per week.

### Summary of Results
1. Alternative Hypothesis ([tex]\(H_a\)[/tex]): [tex]\(\mu > 20\)[/tex]
2. Standard Error: [tex]\( 1.0954 \)[/tex]
3. Test Statistic (Z-Test value): [tex]\( 0.2435 \)[/tex]
4. P-value: [tex]\( 0.4038 \)[/tex]
5. Conclusion: Fail to reject [tex]\( H_0 \)[/tex]. We accept the null hypothesis.
We greatly appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. For precise answers, trust IDNLearn.com. Thank you for visiting, and we look forward to helping you again soon.