Q: A hospital performs 20 liver transplants in a year. The probability of a successful operation is 0.75. What is the probability that exactly 15 patients have a successful operation? Use the binomial formula.

$$P(X=15) = \dbinom{20}{15} \times 0.75$$^{15}$$ \times 0.25$$^{5}$$

Q: A newborn baby is in the 99th percentile for birthweight. If birthweights are normally distributed with mean $$\mu = 7.5$$ pounds and $$\sigma = 1.25$$ pounds, what is the baby's weight?

$$X = \mu + z_{0.99} \times \sigma = 7.5 + 2.326 \times 1.25 = 10.41$$ pounds

Q: Which formula would you use to calculate the probability that at most 15 patients out of 50 have an infectious disease, given $$p=0.25$$?

$$P(X \leq 15) = \text{pbinom}(15, 50, 0.25)$$

Q: A sample of 36 students has a mean of 13 missed school days and a standard deviation of 5. What is the 95% confidence interval for the mean number of missed days?

$$CI = \bar{x} \pm z_{0.975} \times \frac{s}{\sqrt{n}} = 13 \pm 1.96 \times \frac{5}{6}$$

Q: Which R function would you use to find the value $$x$$ such that the area under the normal curve to the left of $$x is 0.90$$, given mean $$\mu$$ and standard deviation $$\sigma$$?

$$\text{qnorm}(0.90, \mu, \sigma)$$

Q: In a binomial experiment with $$n=50$$ and $$p=0.25$$, what is the probability that at least 10 people have the disease?

$$P(X \geq 10) = 1 - \text{pbinom}(9, 50, 0.25)$$

Question 1

Which of the following is an example of a discrete random variable?

Accepted Answer

The number of speeding tickets you have received.

Question 2

A hospital performs 20 liver transplants in a year. The probability of a successful operation is 0.75. What is the probability that exactly 15 patients have a successful operation? Use the binomial formula.

Accepted Answer

$$P(X=15) = \dbinom{20}{15} 	imes 0.75$$^{15}$$ 	imes 0.25$$^{5}$$

Question 3

Which statement best describes the Central Limit Theorem?

Accepted Answer

The distribution of sample means approaches a normal distribution as sample size increases, regardless of the population's distribution.

Question 4

A newborn baby is in the 99th percentile for birthweight. If birthweights are normally distributed with mean $$\mu = 7.5$$ pounds and $$\sigma = 1.25$$ pounds, what is the baby's weight?

Accepted Answer

$$X = \mu + z_{0.99} 	imes \sigma = 7.5 + 2.326 	imes 1.25 = 10.41$$ pounds

Question 5

Which formula would you use to calculate the probability that at most 15 patients out of 50 have an infectious disease, given $$p=0.25$$?

Accepted Answer

$$P(X \leq 15) = 	ext{pbinom}(15, 50, 0.25)$$

Question 6

If the mean SBP (systolic blood pressure) in a sample is 132 mmHg, and the 99% confidence interval is (130, 134), what does this interval mean?

Accepted Answer

We are 99% confident that the true population mean SBP lies between 130 and 134 mmHg.

Question 7

Which of the following is NOT a property of the normal distribution?

Accepted Answer

It is defined only for discrete random variables.

Question 8

A sample of 36 students has a mean of 13 missed school days and a standard deviation of 5. What is the 95% confidence interval for the mean number of missed days?

Accepted Answer

$$CI = \bar{x} \pm z_{0.975} 	imes \frac{s}{\sqrt{n}} = 13 \pm 1.96 	imes \frac{5}{6}$$

Question 9

Which R function would you use to find the value $$x$$ such that the area under the normal curve to the left of $$x is 0.90$$, given mean $$\mu$$ and standard deviation $$\sigma$$?

Accepted Answer

$$	ext{qnorm}(0.90, \mu, \sigma)$$

Question 10

In a binomial experiment with $$n=50$$ and $$p=0.25$$, what is the probability that at least 10 people have the disease?

Accepted Answer

$$P(X \geq 10) = 1 - 	ext{pbinom}(9, 50, 0.25)$$

Question 11

A population has a mean glucose value of $$\mu = 102.5 mg/dl $$and $$\sigma = 28 mg/dl. If $$you take a random sample of $$n=50$$, what is the expected distribution of the sample mean?

Accepted Answer

Normal distribution with mean $$\mu = 102.5$$ and standard error $$\sigma_{\bar{x}} = \frac{28}{\sqrt{50}}$$

Question 12

Which of the following best describes a continuous random variable?

Accepted Answer

A variable that can take any value within a given range, such as height or time.

Question 13

If the probability of success in a binomial experiment is $$p$$, what is the probability of failure?

Accepted Answer

$$1 - p$$

Question 14

Which of the following is true about the parameters of the normal distribution?

Accepted Answer

The mean ($$\mu) $$determines the center, and the standard deviation ($$\sigma) $$determines the spread.

Question 15

A researcher wants to compare the mean BMI between males and females. Which statistical method is most appropriate?

Accepted Answer

Two-sample t-test

Question 16

If a confidence interval for the difference between two group means includes zero, what does this imply?

Accepted Answer

There is no statistically significant difference between the groups at the chosen confidence level.

Question 17

Which R function would you use to calculate the probability that a normally distributed variable is less than a given value $$x$$?

Accepted Answer

$$	ext{pnorm}(x, \mu, \sigma)$$

Question 18

A healthy pregnancy is defined as a birthweight less than 9 pounds. If birthweights are normally distributed with $$\mu = 7.5$$ and $$\sigma = 1.25$$, what percentage of births may require C-section intervention?

Accepted Answer

$$P(X \u003e 9) = 1 - 	ext{pnorm}(9, 7.5, 1.25)$$

Biostatistics Week 3: Random Variables, Binomial and Normal Distributions

Study Guide - Practice Questions

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