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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