Math / Data & Statistics

Question3) According to a study done by the Pew Research Center, 39\% of adult Americans believe that marriage is now obsolete. Suppose a random sample of 500 adult Americans is asked whether marriage is obsolete. When calculating a probability, draw the graph of the normal curve and shade the appropriate area. a. (3 points) Verify the three conditions for the distribution of the sample proportion to be normally distributed. b. (1 point) Calculate the mean and standard deviation for the distribution of the sample proportion. Page 3 of 4
Name: $\qquad$ Score: $\qquad$ /20 pts c. (1 point) What is the probability that in a random sample of 500 adult Americans less than $38 \%$ believe that marriage is obsolete? d. (1 point) What is the probability that in a random sample of 500 adult Americans between $40 \%$ and $45 \%$ believe that marriage is obsolete? e. (1 point) Would it be unusual for a random sample of 500 adult Americans to result in 210 or more who believe marriage is obsolete? Show your work.

Studdy Solution

STEP 1

1. The sample size is $n = 500$ .
2. The population proportion is $p = 0.39$ .
3. We are using the normal approximation for the sample proportion.

STEP 2

1. Verify conditions for normality.
2. Calculate the mean and standard deviation of the sample proportion.
3. Calculate the probability for less than 38%.
4. Calculate the probability for between 40% and 45%.
5. Determine if 210 or more is unusual.

STEP 3

Verify conditions for normality: - Condition 1: The sample is random. - Condition 2: The sample size $n$ is large enough such that $np \geq 10$ and $n(1-p) \geq 10$ . - Condition 3: The sample size is less than 10% of the population.
Calculate: $np = 500 \times 0.39 = 195$ $n(1-p) = 500 \times 0.61 = 305$
Both are greater than 10, and assuming the sample is random and less than 10% of the population, the conditions are satisfied.

STEP 4

Calculate the mean and standard deviation of the sample proportion: - Mean ( $\mu_{\hat{p}}$ ):
$\mu_{\hat{p}} = p = 0.39$
- Standard deviation ( $\sigma_{\hat{p}}$ ):
$\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.39 \times 0.61}{500}} \approx 0.0219$

STEP 5

Calculate the probability that less than 38% believe marriage is obsolete: - Convert 38% to a proportion: $\hat{p} = 0.38$ - Calculate the z-score:
$z = \frac{\hat{p} - \mu_{\hat{p}}}{\sigma_{\hat{p}}} = \frac{0.38 - 0.39}{0.0219} \approx -0.4566$
- Use the standard normal distribution to find the probability:
$P(Z < -0.4566) \approx 0.324$

STEP 6

Calculate the probability that between 40% and 45% believe marriage is obsolete: - Convert 40% and 45% to proportions: $\hat{p}_1 = 0.40$ , $\hat{p}_2 = 0.45$ - Calculate the z-scores:
$z_1 = \frac{0.40 - 0.39}{0.0219} \approx 0.4566$ $z_2 = \frac{0.45 - 0.39}{0.0219} \approx 2.7397$
- Use the standard normal distribution to find the probabilities:
$P(0.4566 < Z < 2.7397) = P(Z < 2.7397) - P(Z < 0.4566) \approx 0.9969 - 0.675 \approx 0.3219$

STEP 7

Determine if 210 or more is unusual: - Convert 210 to a proportion: $\hat{p} = \frac{210}{500} = 0.42$ - Calculate the z-score:
$z = \frac{0.42 - 0.39}{0.0219} \approx 1.3699$
- Use the standard normal distribution to find the probability:
$P(Z > 1.3699) \approx 1 - 0.9147 = 0.0853$
Since $0.0853$ is greater than 0.05, it is not considered unusual.

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