Q: A medical clinic found that 30% of its patients had illnesses of a psychosomatic nature. A psychologist wants to test if this percentage has changed. In a random sample of 80 patients, 35 have psychosomatic illnesses. What is the correct null and alternative hypothesis for this scenario?

$$H_0$$: p = 0.30$$; H_a$$: p \neq 0.30$$

Q: An entrepreneur wants to open a restaurant if more than 15% of local residents are interested. In a survey of 500 people, 90 express interest. What is the correct test statistic to use?

$$z = \frac{\hat{p} - 0.15}{\sqrt{\frac{0.15 \times 0.85}{500}}}$$

Question 1

A medical clinic found that 30% of its patients had illnesses of a psychosomatic nature. A psychologist wants to test if this percentage has changed. In a random sample of 80 patients, 35 have psychosomatic illnesses. What is the correct null and alternative hypothesis for this scenario?

Accepted Answer

$$H_0$$: p = 0.30$$; H_a$$: p 
eq 0.30$$

Question 2

Which of the following best describes a Type I error in hypothesis testing?

Accepted Answer

Concluding that the alternative hypothesis is true when the null hypothesis is actually true.

Question 3

A frozen orange juice producer claims that 20% of all orange juice drinkers prefer its product. A competitor samples 200 drinkers and finds that 34 prefer the producer's brand. What is the value of the sample proportion $$\hat{p}$$?

Accepted Answer

$$\hat{p} = \frac{34}{200} = 0.17$$

Question 4

Which formula is used to calculate the test statistic $$z$$ for testing a population proportion $$p$$?

Accepted Answer

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

Question 5

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

Accepted Answer

The probability of obtaining the observed value of the test statistic, or a value more extreme, assuming the null hypothesis is true.

Question 6

A new cancer screening method is tested on 70 patients known to have cancer. It detects cancer in 64 patients. The current method detects cancer in 85% of cases. What is the sample proportion for the new method?

Accepted Answer

$$\hat{p} = \frac{64}{70} \approx 0.914$$

Question 7

If the significance level $$\alpha is $$set to 0.05, what is the decision rule for a hypothesis test?

Accepted Answer

Reject $$H_0 if $$the p-value is less than 0.05.

Question 8

An entrepreneur wants to open a restaurant if more than 15% of local residents are interested. In a survey of 500 people, 90 express interest. What is the correct test statistic to use?

Accepted Answer

$$z = \frac{\hat{p} - 0.15}{\sqrt{\frac{0.15 	imes 0.85}{500}}}$$

Question 9

Which of the following is NOT an element of a test of hypothesis?

Accepted Answer

Calculation of the sample mean

Question 10

A consumer group suspects that less than 60% of graduate students have purchased online. In a sample of 80 students, 42 have purchased online. What is the sample proportion?

Accepted Answer

$$\hat{p} = \frac{42}{80} = 0.525$$

Question 11

Which of the following statements about Type II error is correct?

Accepted Answer

It is the probability of failing to reject the null hypothesis when the alternative hypothesis is true.

Question 12

A sample of 2000 orange juice drinkers finds that 17% prefer the producer's brand. If $$\alpha = 0.10$$, what effect does increasing the sample size have on the hypothesis test?

Accepted Answer

It decreases the standard error, making it easier to detect a significant difference.

Question 13

Which formula gives the 95% confidence interval for a population proportion $$p$$?

Accepted Answer

$$\hat{p} \pm z_{0.025} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$$

Question 14

In the context of hypothesis testing, what does the significance level $$\alpha$$ represent?

Accepted Answer

The probability of making a Type I error.

Question 15

A test of hypothesis about a population proportion $$p is $$appropriate when:

Accepted Answer

The population of interest contains qualitative data and the parameter of interest is $$p.$$

Question 16

If a researcher sets $$\alpha = 0.01$$ for a test, what is the most likely reason?

Accepted Answer

The consequences of a Type I error are severe and should be minimized.

Question 17

Which of the following is a correct interpretation of a p-value of 0.03 in a test with $$\alpha = 0.05$$?

Accepted Answer

There is sufficient evidence to reject the null hypothesis at the 5% significance level.

Question 18

A test statistic $$z is $$calculated as $$z = -2.5. If $$the test is two-tailed and $$\alpha = 0.05$$, what should you conclude?

Accepted Answer

Reject the null hypothesis if the p-value corresponding to $$|z| = 2.5 is $$less than 0.05.

Question 19

Which of the following scenarios best illustrates a Type II error?

Accepted Answer

Failing to detect a new drug's effectiveness when it actually works.

Hypothesis Testing for Population Proportions

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