Questions: Average Family Size The average family size was reported as 3.18. A random sample of families in a particular school district resulted in the following family sizes: 3 2 7 5 4 3 6 2 5 4 3 2 7 4 2 5 3 3 2 3 5 2 9 8 At a=0.05, does the average family size differ from the national average? Assume the population is normally distributed. Part 1 of 5 (a) State the hypotheses and identify the claim. H0: μ=3.18 not claim H1: μ ≠ 3.18 claim This hypothesis test is a two-tailed test. Part 2 of 5 (b) Find the critical value(s). Round the answer to three decimal places, if necessary. If there is more than one critical value, separate them with commas. Critical value(s): 2.069,-2.069

Average Family Size The average family size was reported as 3.18. A random sample of families in a particular school district resulted in the following family sizes: 3 2 7 5 4 3 6 2 5 4 3 2 7 4 2 5 3 3 2 3 5 2 9 8 At a=0.05, does the average family size differ from the national average? Assume the population is normally distributed. Part 1 of 5 (a) State the hypotheses and identify the claim. H0: μ=3.18 not claim H1: μ ≠ 3.18 claim This hypothesis test is a two-tailed test. Part 2 of 5 (b) Find the critical value(s). Round the answer to three decimal places, if necessary. If there is more than one critical value, separate them with commas. Critical value(s): 2.069,-2.069
Transcript text: Average Family Size The average family size was reported as 3.18. A random sample of families in a particular school district resulted in the following family sizes: 3 2 7 5 4 3 6 2 5 4 3 2 7 4 2 5 3 3 2 3 5 2 9 8 At $a=0.05$, does the average family size differ from the national average? Assume the population is normally distributed. Part 1 of 5 (a) State the hypotheses and identify the claim. \[ \begin{array}{l} H_{0}: \mu=3.18 \text { not claim } \\ H_{1}: \mu \neq 3.18 \text { claim } \end{array} \] This hypothesis test is a two-tailed test. Part 2 of 5 (b) Find the critical value(s). Round the answer to three decimal places, if necessary. If there is more than one critical value, separate them with commas. \[ \text { Critical value(s): } 2.069,-2.069 \]
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Solution

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Solution Steps

Step 1: State the Hypotheses

We are testing whether the average family size differs from the national average of 3.18. The hypotheses are stated as follows:

\[ \begin{align_} H_{0}: & \quad \mu = 3.18 \quad \text{(not claim)} \\ H_{1}: & \quad \mu \neq 3.18 \quad \text{(claim)} \end{align_} \]

This is a two-tailed test.

Step 2: Calculate Sample Statistics

From the sample data, we calculated the following statistics:

  • Sample Size (\(n\)): 24
  • Sample Mean (\(\bar{x}\)): 4.12
  • Sample Standard Deviation (\(s\)): 2.05

The Standard Error (\(SE\)) is calculated as:

\[ SE = \frac{s}{\sqrt{n}} = \frac{2.0497}{\sqrt{24}} \approx 0.4184 \]

Step 3: Compute the Test Statistic

The test statistic \(t\) is calculated using the formula:

\[ t = \frac{\bar{x} - \mu_0}{SE} = \frac{4.125 - 3.18}{0.4184} \approx 2.2587 \]

Rounding to three decimal places, we have:

\[ t \approx 2.259 \]

Step 4: Calculate the P-value

For a two-tailed test, the P-value is calculated as:

\[ P = 2 \times (1 - T(|t|)) \approx 0.0337 \]

Final Answer

The test statistic is \(t \approx 2.259\) and the P-value is approximately \(0.0337\).

Thus, we conclude:

\[ \boxed{t \approx 2.259} \]

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