Q: A sample of 25 patients has a mean cholesterol level of 190 mg/dL and a standard deviation of 20 mg/dL. What is the 95% confidence interval for the mean cholesterol level? (Assume normality and use $$t_{0.025,24} \approx 2.064)$$

$$190 \pm 2.064 \times \frac{20}{\sqrt{25}}$$

Question 1

Given a dataset with values: 2, 4, 4, 4, 5, 5, 7, 9, which measure of central tendency is most affected by the presence of the value 9?

Accepted Answer

The mean

Question 2

A pharmaceutical company tests a new drug on 100 patients and records the time (in days) until symptoms disappear. If the data is heavily right-skewed, which measure of central tendency should be used to best represent the typical recovery time?

Accepted Answer

The median

Question 3

Which formula correctly calculates the sample variance $$s^2$$ for a dataset $$x_1$$, $$x_2$$, ..., x_n$$ with sample mean $$\bar{x}$$?

Accepted Answer

$$s^2$$ = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x}$$)^2$$

Question 4

A hospital records the blood pressure of 200 patients. The mean systolic blood pressure is 120 mmHg with a standard deviation of 15 mmHg. What percentage of patients are expected to have systolic blood pressure between 105 mmHg and 135 mmHg, assuming a normal distribution?

Accepted Answer

Approximately 68%

Question 5

A researcher wants to estimate the proportion of students who prefer online classes. She surveys 40 students and finds that 28 prefer online classes. What is the sample proportion $$\hat{p}$$?

Accepted Answer

$$\hat{p} = \frac{28}{40} = 0.7$$

Question 6

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

Accepted Answer

It is skewed to the right

Question 7

A factory produces light bulbs with lifespans that are normally distributed with mean 800 hours and standard deviation 60 hours. What is the $$z-$$score for a bulb that lasts 920 hours?

Accepted Answer

$$z = \frac{920 - 800}{60} = 2$$

Question 8

A medical test for a disease has a sensitivity of 95% and a specificity of 90%. If the disease prevalence is 2%, what is the probability that a randomly selected person who tests positive actually has the disease? (Use Bayes' theorem.)

Accepted Answer

Approximately 16.1%

Question 9

Which of the following best describes the Central Limit Theorem?

Accepted Answer

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

Question 10

A study finds that the correlation coefficient between hours studied and exam score is $$r = 0.85. $$What does this imply?

Accepted Answer

There is a strong positive linear relationship between hours studied and exam score.

Question 11

A sample of 25 patients has a mean cholesterol level of 190 mg/dL and a standard deviation of 20 mg/dL. What is the 95% confidence interval for the mean cholesterol level? (Assume normality and use $$t_{0.025,24} \approx 2.064)$$

Accepted Answer

$$190 \pm 2.064 	imes \frac{20}{\sqrt{25}}$$

Question 12

Which of the following statements about $$p-$$values is correct?

Accepted Answer

A $$p-$$value is the probability of observing data as extreme as the sample, assuming the null hypothesis is true.

Question 13

A researcher performs a hypothesis test and obtains a $$p-$$value of 0.03. If the significance level is $$\alpha = 0.05$$, what is the correct conclusion?

Accepted Answer

Reject the null hypothesis

Question 14

Which formula gives the standard error of the mean for a sample of size $$n$$ with standard deviation $$s$$?

Accepted Answer

$$SE = \frac{s}{\sqrt{n}}$$

Question 15

A university wants to compare the average GPA of students in two different majors. Which statistical test is most appropriate?

Accepted Answer

Two-sample $$t-$$test

Question 16

A survey asks respondents to rate their satisfaction on a scale from 1 to 5. What type of data is this?

Accepted Answer

Ordinal

Question 17

Which formula is used to calculate the expected value $$E[X] of a $$discrete random variable $$X$$ with possible values $$x_i$$ and probabilities $$p_i$$?

Accepted Answer

$$E[X] = \sum_{i} x_i p_i$$

Question 18

A clinical trial compares the effectiveness of three different treatments for hypertension. Which statistical test should be used to determine if there are significant differences among the treatments?

Accepted Answer

ANOVA

Question 19

A researcher collects data on the number of cars passing through an intersection each hour. Which probability distribution is most appropriate to model this data?

Accepted Answer

Poisson distribution

Sampling Distributions and Confidence Intervals for Proportions: Study Notes

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