Q: A pharmaceutical company claims that its new drug cures 80% of patients. In a clinical trial of 50 patients, 35 are cured. What is the value of the test statistic for testing the company's claim? (Use $$z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}$$ and $$\mu = 0.8$$, $$\bar{x} = 0.7$$, $$\sigma = \sqrt{0.8 \times 0.2})$$

$$z = \frac{0.7 - 0.8}{\sqrt{0.8 \times 0.2} / \sqrt{50}}$$

Q: A quality control manager wants to estimate the proportion of defective items produced by a machine. She wants the margin of error to be at most 0.02 with 95% confidence. If the estimated proportion is 0.1, what is the minimum sample size required? (Use $$z_c = 1.96)$$

$$n = \left( \frac{1.96 \sqrt{0.1 \times 0.9}}{0.02} \$$right)^2$$

Question 1

Which of the following statements best distinguishes a parameter from a statistic?

Accepted Answer

A parameter is a descriptive measure of a population, while a statistic is a descriptive measure of a sample.

Question 2

A researcher wants to estimate the average cholesterol level in adults in a city. She randomly selects 100 adults and finds their mean cholesterol level is 190 mg/dL. Which of the following is the best point estimate for the population mean?

Accepted Answer

190 mg/dL

Question 3

Suppose the heights of adult males in a country are normally distributed with mean $$\mu = 70$$ inches and standard deviation $$\sigma = 3$$ inches. What is the z-score for a man who is 76 inches tall?

Accepted Answer

$$z = \frac{76 - 70}{3} = 2$$

Question 4

A fair coin is flipped 10 times. What is the probability of getting exactly 6 heads? (Use the binomial formula: $$P(r) = \binom{n}{r} p^r q$$^{n-r}$$, where p=0.5$ and q=0.5$)

Accepted Answer

$$P(6) = \binom{10}{6} (0.5)^6$ (0.5)^4$

Question 5

Which of the following is NOT an assumption for using the confidence interval formula $$\bar{x} \pm z_c \left( \frac{\sigma}{\sqrt{n}} \right)$$ for the population mean $$\mu$$?

Accepted Answer

The population standard deviation $$\sigma is $$unknown.

Question 6

A sample of 25 students has a mean test score of 80 and a standard deviation of 10. Construct a 95% confidence interval for the population mean, assuming the population is normal and $$\sigma is $$unknown. (Use $$t_{c} \approx 2.064$$ for $$df=24)$$

Accepted Answer

$$80 \pm 2.064 \left( \frac{10}{\sqrt{25}} \right)$$

Question 7

Which of the following best describes the Central Limit Theorem?

Accepted Answer

For any population, the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the population's distribution.

Question 8

A pharmaceutical company claims that its new drug cures 80% of patients. In a clinical trial of 50 patients, 35 are cured. What is the value of the test statistic for testing the company's claim? (Use $$z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}$$ and $$\mu = 0.8$$, $$\bar{x} = 0.7$$, $$\sigma = \sqrt{0.8 	imes 0.2})$$

Accepted Answer

$$z = \frac{0.7 - 0.8}{\sqrt{0.8 	imes 0.2} / \sqrt{50}}$$

Question 9

Which of the following is a correct interpretation of a 95% confidence interval for the mean weight of apples in an orchard?

Accepted Answer

We are 95% confident that the interval contains the true mean weight of apples in the orchard.

Question 10

A quality control manager wants to estimate the proportion of defective items produced by a machine. She wants the margin of error to be at most 0.02 with 95% confidence. If the estimated proportion is 0.1, what is the minimum sample size required? (Use $$z_c = 1.96)$$

Accepted Answer

$$n = \left( \frac{1.96 \sqrt{0.1 	imes 0.9}}{0.02} \$$right)^2$$

Question 11

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

Accepted Answer

Rejecting the null hypothesis when it is actually true.

Question 12

A researcher is testing whether a new teaching method improves test scores. The null hypothesis is $$H_0$$: \mu = 75$$ and the alternative is H_1$: \mu \u003e 75. $$Which type of test should be used?

Accepted Answer

Right-tailed test

Question 13

If the probability of event A is 0.3 and the probability of event B is 0.5, and A and B are mutually exclusive, what is the probability that either A or B occurs?

Accepted Answer

$$P(A \cup B) = 0.3 + 0.5 = 0.8$$

Question 14

A discrete probability distribution is valid if:

Accepted Answer

Each probability is between 0 and 1, and the sum of all probabilities is exactly 1.

Question 15

Which of the following is a feature of the t-distribution compared to the standard normal (z) distribution?

Accepted Answer

The t-distribution has thicker tails than the z-distribution.

Question 16

A normal approximation to the binomial distribution can be used if:

Accepted Answer

Both $$np \u003e 5$$ and $$nq \u003e 5$$

Question 17

A sample of 36 patients has a mean blood pressure of 120 mmHg. The population standard deviation is known to be 12 mmHg. What is the standard error of the mean?

Accepted Answer

$$\frac{12}{\sqrt{36}} = 2$$

Question 18

If the mean of a binomial distribution is $$\mu = np$$ and the standard deviation is $$\sigma = \sqrt{npq}$$, what is the mean and standard deviation for 100 trials with $$p = 0.4$$?

Accepted Answer

$$\mu = 100 	imes 0.4 = 40$$, $$\sigma = \sqrt{100 	imes 0.4 	imes 0.6} = 4.898$$

Question 19

A researcher calculates a z-score of 2.5 for a sample mean. For a right-tailed test at $$\alpha = 0.01$$, what is the correct conclusion?

Accepted Answer

Reject $$H_0$$ because the P-value is less than $$\alpha.$$

Comprehensive Study Notes for Introductory Statistics (Chapters 1–7)

Guida di studio - Domande di pratica

Study guide - Flashcard