Questions: Depression and New Deal Great War and Jazz Age Sections 7.4 7.6 HW: The Central Limit Theorem for Proportions and Assessing Normality Question 3 of 13 (5 points) I Question Attempt: 1 of Unlimited Below, n is the sample size, p is the population proportion, and p̂ is the sample proportion. First, check if the assumptions are satisfied to use the n distribution for probabilities. If appropriate, use the Central Limit Theorem to find the indicated probability. n=147 p=0.17 Part 1 of 2 It (Choose one) appropriate to use the normal distribution for probabilities. Part 2 of 2 P(0.15<p̂<0.19)=

Depression and New Deal Great War and Jazz Age

Sections 7.4 7.6 HW: The Central Limit Theorem for Proportions and Assessing Normality Question 3 of 13 (5 points) I Question Attempt: 1 of Unlimited

Below, n is the sample size, p is the population proportion, and p̂ is the sample proportion. First, check if the assumptions are satisfied to use the n distribution for probabilities. If appropriate, use the Central Limit Theorem to find the indicated probability.

n=147
p=0.17

Part 1 of 2

It (Choose one) appropriate to use the normal distribution for probabilities.

Part 2 of 2
P(0.15<p̂<0.19)=

Transcript text: Depression and New Deal Great War and Jazz Age Sections 7.4 & 7.6 HW: The Central Limit Theorem for Proportions and Assessing Normality Question 3 of 13 ( 5 points) I Question Attempt: 1 of Unlimited Below, $n$ is the sample size, $p$ is the population proportion, and $\hat{p}$ is the sample proportion. First, check if the assumptions are satisfied to use the $n$ distribution for probabilities. If appropriate, use the Central Limit Theorem to find the indicated probability. \[ \begin{array}{l} n=147 \\ p=0.17 \end{array} \] Part 1 of 2 It (Choose one) \ appropriate to use the normal distribution for probabilities. $\square$ Part 2 of 2 $P(0.15<\hat{p}<0.19)=$ $\square$ $\square$

Solution

Solution Steps

To determine if it's appropriate to use the normal distribution for probabilities, we need to check the conditions for the Central Limit Theorem (CLT) for proportions. Specifically, we need to ensure that both $ n \times p $ and $ n \times (1-p) $ are greater than 5. If these conditions are met, we can use the normal distribution to approximate the sampling distribution of the sample proportion $\hat{p}$. Then, we calculate the probability $ P(0.15 < \hat{p} < 0.19) $ using the normal distribution with mean $ p $ and standard deviation $\sqrt{\frac{p(1-p)}{n}}$.

Step 1: Verify Conditions for Normal Approximation

To determine if it is appropriate to use the normal distribution for probabilities, we check the conditions for the Central Limit Theorem (CLT) for proportions. Specifically, we need to ensure that both $ n \times p $ and $ n \times (1-p) $ are greater than 5.

Given:

$ n = 147 $
$ p = 0.17 $

Calculate:

$ n \times p = 147 \times 0.17 = 24.99 $
$ n \times (1-p) = 147 \times (1 - 0.17) = 122.01 $

Both values are greater than 5, so it is appropriate to use the normal distribution.

Step 2: Calculate Mean and Standard Deviation

The mean of the sampling distribution of the sample proportion $\hat{p}$ is given by: \[ \text{mean} = p = 0.17 \]

The standard deviation is given by: \[ \text{std\_dev} = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.17 \times (1-0.17)}{147}} \approx 0.0310 \]

Step 3: Calculate the Probability $ P(0.15 < \hat{p} < 0.19) $

Using the normal distribution with the calculated mean and standard deviation, we find: \[ P(0.15 < \hat{p} < 0.19) = P(\hat{p} < 0.19) - P(\hat{p} < 0.15) \]

The probability is approximately: \[ P(0.15 < \hat{p} < 0.19) \approx 0.4814 \]