Q: Given a binomial distribution with $$n = 20$$ and $$p = 0.3$$, what is the mean ($$\mu) of $$the distribution?

$$\mu = n \times p = 20 \times 0.3 = 6$$

Q: For a binomial distribution with $$n = 15$$ and $$p = 0.4$$, what is the variance ($$\$$sigma^2$$)?

$$\$$sigma^2$$ = n \times p \times (1-p) = 15 \times 0.4 \times 0.6 = 3.6$$

Q: A researcher wants to estimate the mean cholesterol level in adults. She collects a sample of 50 adults, finds a sample mean of 200 mg/dL, and a sample standard deviation of 20 mg/dL. What is the margin of error for a 95% confidence interval?

Margin of error $$= z_{0.025} \times \frac{\sigma}{\sqrt{n}} = 1.96 \times \frac{20}{\sqrt{50}} \approx 5.54$$

Q: A sample of 25 students has a mean test score of 80 and a standard deviation of 10. What is the 90% confidence interval for the population mean?

CI: $$80 \pm t_{0.05,24} \times \frac{10}{\sqrt{25}} = 80 \pm 1.711 \times 2 = 80 \pm 3.422$$; interval: $$(76.578, 83.422)$$

Q: A researcher wants to find the sample size needed to estimate the mean systolic blood pressure within 4 mmHg with 95% confidence, assuming $$\sigma = 12$$ mmHg. What is the required sample size?

$$n = \left(\frac{z_{0.025} \times \sigma}{E}\$$right)^2$$ = \left(\frac{1.96 \times 12}{4}\$$right)^2$$ = (5.88)^2$ = 34.574 \approx 35$$

Question 1

Which of the following is NOT a requirement for a probability distribution?

Accepted Answer

The random variable must be continuous.

Question 2

A pharmaceutical company tests a new drug on 100 patients and records whether each patient recovers (success) or not (failure). Which condition confirms this is a binomial experiment?

Accepted Answer

Each trial has only two possible outcomes.

Question 3

Given a binomial distribution with $$n = 20$$ and $$p = 0.3$$, what is the mean ($$\mu) of $$the distribution?

Accepted Answer

$$\mu = n 	imes p = 20 	imes 0.3 = 6$$

Question 4

For a binomial distribution with $$n = 15$$ and $$p = 0.4$$, what is the variance ($$\$$sigma^2$$)?

Accepted Answer

$$\$$sigma^2$$ = n 	imes p 	imes (1-p) = 15 	imes 0.4 	imes 0.6 = 3.6$$

Question 5

A hospital records the weights of newborns, which are normally distributed with mean $$\mu = 3.5 kg $$and standard deviation $$\sigma = 0.5 kg. $$What is the probability that a randomly selected newborn weighs less than 3 kg?

Accepted Answer

Find $$z = \frac{3 - 3.5}{0.5} = -1. $$The probability is $$P(X \u003c 3) = P(z \u003c -1) \approx 0.1587$$

Question 6

A standardized test has scores that are normally distributed with $$\mu = 500$$ and $$\sigma = 100. $$What is the probability that a randomly selected student scores between 400 and 600?

Accepted Answer

Find $$z_1$$ = \frac{400 - 500}{100} = -1$$, z_2$ = \frac{600 - 500}{100} = 1$$. $$P(400 \u003c X \u003c 600) = P(-1 \u003c z \u003c 1) \approx 0.6826$$

Question 7

A researcher wants to estimate the mean cholesterol level in adults. She collects a sample of 50 adults, finds a sample mean of 200 mg/dL, and a sample standard deviation of 20 mg/dL. What is the margin of error for a 95% confidence interval?

Accepted Answer

Margin of error $$= z_{0.025} 	imes \frac{\sigma}{\sqrt{n}} = 1.96 	imes \frac{20}{\sqrt{50}} \approx 5.54$$

Question 8

A sample of 25 students has a mean test score of 80 and a standard deviation of 10. What is the 90% confidence interval for the population mean?

Accepted Answer

CI: $$80 \pm t_{0.05,24} 	imes \frac{10}{\sqrt{25}} = 80 \pm 1.711 	imes 2 = 80 \pm 3.422$$; interval: $$(76.578, 83.422)$$

Question 9

A population has mean $$\mu = 100$$ and standard deviation $$\sigma = 15. $$What is the mean and standard deviation of the sampling distribution of the sample mean for samples of size $$n = 36$$?

Accepted Answer

Mean $$= 100$$, Standard deviation $$= \frac{15}{\sqrt{36}} = 2.5$$

Question 10

A researcher wants to find the sample size needed to estimate the mean systolic blood pressure within 4 mmHg with 95% confidence, assuming $$\sigma = 12$$ mmHg. What is the required sample size?

Accepted Answer

$$n = \left(\frac{z_{0.025} 	imes \sigma}{E}\$$right)^2$$ = \left(\frac{1.96 	imes 12}{4}\$$right)^2$$ = (5.88)^2$ = 34.574 \approx 35$$

Question 11

A student scores 85 on a test with mean 75 and standard deviation 5. Another student scores 90 on a test with mean 80 and standard deviation 10. Who performed better relative to their group?

Accepted Answer

First student: $$z = \frac{85-75}{5} = 2$$; Second student: $$z = \frac{90-80}{10} = 1. $$The first student performed better.

Question 12

A clinical trial compares the effect of two drugs on blood pressure. Each patient receives both drugs at different times. Are the samples dependent or independent?

Accepted Answer

Dependent, because the same subjects are measured under both conditions.

Question 13

Which scenario requires the use of a t-distribution rather than a z-distribution?

Accepted Answer

When the population standard deviation is unknown and the sample size is small ($$n \u003c 30).$$

Question 14

In hypothesis testing, what is a Type I error?

Accepted Answer

Rejecting the null hypothesis when it is actually true.

Question 15

A researcher tests whether a new fertilizer increases plant growth. The null hypothesis is $$H_0$$: \mu = 10$$ cm. The sample mean is 12 cm, and the 95% confidence interval is (11, 13). What should the researcher conclude?

Accepted Answer

Reject $$H_0$$ because 10 cm is not within the confidence interval.

Question 16

A hypothesis test has a significance level $$\alpha = 0.05. $$The p-value is 0.03. What is the correct decision?

Accepted Answer

Reject the null hypothesis because $$p \u003c \alpha.$$

Question 17

A test statistic is calculated as $$z = 2.1$$ for a right-tailed test at $$\alpha = 0.05. $$What is the critical value and what is the correct decision?

Accepted Answer

Critical value $$z_{0.05} = 1.645. $$Since $$z = 2.1 \u003e 1.645$$, reject the null hypothesis.

Question 18

A medical study tests whether a new treatment is more effective than the standard. The alternative hypothesis is $$H_1$$: \mu \u003e \$$mu_0. Is $$this a one-tailed or two-tailed test?

Accepted Answer

One-tailed test (right-tailed).

Question 19

A random variable $$X$$ has mean $$\mu = 50$$ and standard deviation $$\sigma = 8. $$What value of $$X$$ corresponds to the 90th percentile?

Accepted Answer

90th percentile $$z \approx 1.28$$. $$X = \mu + z \sigma = 50 + 1.28 	imes 8 = 60.24$$

Midterm 2 Study Guide: Probability Distributions, Normal Distributions, and Inferential Statistics

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