Question 1

Which condition must be satisfied to use the normal approximation for the binomial distribution of sample proportions?

Accepted Answer

Both $$np \geq 5$$ and $$nq \geq 5$$

Question 2

In hypothesis testing for a population proportion, what does $$p$$ represent in the formulas for mean and standard deviation?

Accepted Answer

The assumed proportion in the claim

Question 3

Given a sample size $$n = 220$$ and a sample proportion $$\hat{p} = \frac{91}{220}$$, what is the null hypothesis for testing if the cancer rate among cast and crew is greater than 40%?

Accepted Answer

$$H_0$$: p = 0.40$$

Question 4

What is the formula for the standard deviation of the sampling distribution of sample proportions when using the normal approximation?

Accepted Answer

$$\sigma = \sqrt{\frac{pq}{n}}$$

Question 5

A Pew Research Center survey found that 52% of 926 Internet users utilize two-factor authentication. What is the test statistic for testing the claim that most Internet users use two-factor authentication?

Accepted Answer

$$z = \frac{\hat{p} - p}{\sqrt{\frac{pq}{n}}}$$

Question 6

If the significance level is $$\alpha = 0.05$$ and the test is right-tailed, what is the critical value for the $$z-$$test?

Accepted Answer

$$z = 1.645$$

Question 7

Why is the exact method for hypothesis testing sometimes preferred over the normal approximation?

Accepted Answer

It provides more accurate results, especially for small sample sizes or extreme proportions.

Question 8

In the 'Cursed Movie' scenario, what is the alternative hypothesis for testing if the cancer rate among cast and crew is greater than 40%?

Accepted Answer

$$H_1$$: p \u003e 0.40$$

Question 9

If $$np = 10$$ and $$nq = 12$$, what can be concluded about the sampling distribution of sample proportions?

Accepted Answer

It can be approximated by a normal distribution.

Question 10

A researcher wants to test the claim that the proportion of people using two-factor authentication is greater than 0.5. Which of the following is the correct form of the alternative hypothesis?

Accepted Answer

$$H_1$$: p \u003e 0.5$$

Question 11

If a sample proportion $$\hat{p} is $$much higher than the assumed proportion $$p in $$the null hypothesis, and the calculated $$z-$$score exceeds the critical value, what should you conclude?

Accepted Answer

Reject the null hypothesis.

Question 12

Which real-world scenario best illustrates the use of hypothesis testing for a population proportion?

Accepted Answer

Testing if the rate of cancer among movie cast exposed to radiation is higher than the national average.

Question 13

What is the primary purpose of using a significance level $$\alpha in $$hypothesis testing?

Accepted Answer

To determine the probability of rejecting the null hypothesis when it is true.

Question 14

If the sample size is large and the sample proportion is close to the assumed proportion, what is likely about the $$z-$$score?

Accepted Answer

It will be close to zero.

Question 15

Which of the following is NOT a plausible reason for failing to reject the null hypothesis in a population proportion test?

Accepted Answer

The sample size is too small to detect a difference.

Question 16

In the context of hypothesis testing, what does a p-value represent?

Accepted Answer

The probability of obtaining a result at least as extreme as the observed one, assuming the null hypothesis is true.

Question 17

Which formula would you use to calculate the mean of the sampling distribution of sample proportions?

Accepted Answer

$$\mu = p$$

Question 18

A hypothesis test for a population proportion is conducted using the normal approximation. Which assumption is most critical for the validity of this approximation?

Accepted Answer

Both $$np \geq 5$$ and $$nq \geq 5$$

Question 19

If a hypothesis test concludes that the cancer rate among cast and crew is significantly higher than 40%, what does this imply about the movie being a curse?

Accepted Answer

There is statistical evidence supporting the claim that the movie is a curse.

Hypothesis Testing for Population Proportions – Step-by-Step Guidance

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