Q: Given $p = 0.09$ and $n = 1000$, what is the standard error (SE) of the sample proportion $\hat{p}$?

$SE = \sqrt{\frac{0.09 \times 0.91}{1000}}$

Q: For the restaurant scenario, what is the standard error of the sample proportion of vegans, given $p = 0.04$ and $n = 300$?

$SE = \sqrt{\frac{0.04 \times 0.96}{300}}$

Question 1

A random sample of 1000 letters is taken from a large book, and the percentage of 't's is recorded. If the true proportion of 't's in the book is $p = 0.09$, what is the expected sample percentage of 't's?

Accepted Answer

$9\%$

Question 2

Given $p = 0.09$ and $n = 1000$, what is the standard error (SE) of the sample proportion $\hat{p}$?

Accepted Answer

$SE = \sqrt{\frac{0.09 	imes 0.91}{1000}}$

Question 3

Calculate the numerical value of the standard error for the sample proportion of 't's, given $p = 0.09$ and $n = 1000$.

Accepted Answer

$0.009$

Question 4

If the sample proportion of 't's is $\hat{p} = 0.081$ in a sample of 1000 letters, what is the $z$-score for this sample proportion? (Given $p = 0.09$, $SE = 0.009$)

Accepted Answer

$z = \frac{0.081 - 0.09}{0.009} = -1$

Question 5

What is the probability that a random sample of 1000 letters will contain $8.1\%$ or fewer 't's? (Given $p = 0.09$, $SE = 0.009$)

Accepted Answer

$0.1611$

Question 6

A restaurant expects $4\%$ of its patrons to be vegan. If 300 patrons are expected, what is the expected number of vegan patrons?

Accepted Answer

$12$

Question 7

For the restaurant scenario, what is the standard error of the sample proportion of vegans, given $p = 0.04$ and $n = 300$?

Accepted Answer

$SE = \sqrt{\frac{0.04 	imes 0.96}{300}}$

Question 8

If the restaurant wants to know the probability that 15 or more patrons will be vegan, what is the sample proportion $\hat{p}$ corresponding to 15 vegans out of 300 patrons?

Accepted Answer

$\hat{p} = \frac{15}{300} = 0.05$

Question 9

Calculate the $z$-score for observing $\hat{p} = 0.05$ vegans in a sample of 300, given $p = 0.04$ and $SE = 0.0113$.

Accepted Answer

$z = \frac{0.05 - 0.04}{0.0113} \approx 0.88$

Question 10

What is the probability that 15 or more patrons will be vegan at the restaurant? (Given $z = 0.88$)

Accepted Answer

$0.1894$

Question 11

Which of the following is the correct formula for the standard error of a sample proportion?

Accepted Answer

$SE = \sqrt{\frac{p(1-p)}{n}}$

Question 12

If the sample size increases, what happens to the standard error of the sample proportion, assuming $p$ is constant?

Accepted Answer

The standard error decreases.

Question 13

A sample proportion $\hat{p}$ is considered approximately normal if which of the following conditions are met?

Accepted Answer

$np \geq 10$ and $n(1-p) \geq 10$

Question 14

In the context of the restaurant scenario, what does the area to the right of $z = 0.88$ represent?

Accepted Answer

The probability that 15 or more patrons will be vegan.

Question 15

If a sample of 1000 letters yields a sample proportion of 't's of $0.081$, and the population proportion is $0.09$, what is the interpretation of a $z$-score of $-1$?

Accepted Answer

The sample proportion is one standard error below the population proportion.

Question 16

Suppose the probability of a sample proportion being less than a certain value is $0.1611$. What is the corresponding $z$-score?

Accepted Answer

$-1$

Question 17

If the true proportion of vegans is $0.04$ and a sample of 300 patrons is taken, what is the probability that the sample proportion of vegans is at least $0.05$?

Accepted Answer

$0.1894$

Question 18

Which of the following best describes the Central Limit Theorem as applied to sample proportions?

Accepted Answer

For large $n$, the sampling distribution of $\hat{p}$ is approximately normal, regardless of the population distribution.

Sampling Distributions and Probability of Sample Proportions

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