Q: Suppose the sample mean $$ar{x} is $$calculated from a random sample of size $$n$$ from a population with mean $$\\mu$$ and variance $$\\$$sigma^2$$. What is the sampling distribution of $$ar{x}$$?

It is a normal distribution with mean $$\\mu$$ and variance $$\\$$sigma^2$$/n if $$the population is normal or $$n is $$large.

Q: Which of the following is the correct formula for a $$100(1-\\alpha)\\%$$ confidence interval for a population mean when the population standard deviation is unknown?

$$\\bar{x} \\pm t_{\\alpha/2, n-1} \\cdot \\frac{s}{\\sqrt{n}}$$

Q: Which formula represents the joint probability of two independent events $$A$$ and $$B$$?

$$P(A \\cap B) = P(A) \\cdot P(B)$$

Q: Given the following probabilities: $$P(A) = 0.3$$, $$P(B) = 0.5$$, and $$P(A \\cap B) = 0.15. $$Are events $$A$$ and $$B$$ independent?

Yes, because $$P(A \\cap B) = P(A) \\cdot P(B)$$

Question 1

Which of the following visualizations is most appropriate for displaying the distribution of a continuous variable, such as the heights of students in a class?

Accepted Answer

Histogram

Question 2

A boxplot of cholesterol levels in a population shows a long tail to the right. What does this indicate about the distribution?

Accepted Answer

The distribution is positively skewed.

Question 3

Why might a statistician choose to apply a logarithmic transformation to a dataset before analysis?

Accepted Answer

To make the data more symmetric and reduce skewness.

Question 4

Which of the following best describes the relationship between a population parameter and a point estimator?

Accepted Answer

A point estimator is a statistic calculated from sample data used to estimate the population parameter.

Question 5

Suppose the sample mean $$ar{x} is $$calculated from a random sample of size $$n$$ from a population with mean $$\mu$$ and variance $$\$$sigma^2$$. What is the sampling distribution of $$ar{x}$$?

Accepted Answer

It is a normal distribution with mean $$\mu$$ and variance $$\$$sigma^2$$/n if $$the population is normal or $$n is $$large.

Question 6

What is the significance of the Central Limit Theorem in statistics?

Accepted Answer

It allows us to use the normal distribution to approximate the sampling distribution of the sample mean for large sample sizes, regardless of the population's distribution.

Question 7

A researcher wants to estimate the mean blood pressure in a population. The sample size is small ($$n = 12$$), and the population standard deviation is unknown. Which distribution should be used to construct the confidence interval?

Accepted Answer

t distribution

Question 8

Which of the following is the correct formula for a $$100(1-\alpha)\%$$ confidence interval for a population mean when the population standard deviation is unknown?

Accepted Answer

$$\bar{x} \pm t_{\alpha/2, n-1} \cdot \frac{s}{\sqrt{n}}$$

Question 9

Which statement best describes the interpretation of a 95% confidence interval for the mean weight of newborns in a hospital?

Accepted Answer

If we repeat the sampling process many times, about 95% of the calculated intervals will contain the true mean.

Question 10

How does increasing the confidence level from 90% to 99% affect the width of a confidence interval for a population mean?

Accepted Answer

The width increases.

Question 11

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

Accepted Answer

Rejecting a true null hypothesis.

Question 12

A pharmaceutical company wants to test if a new drug lowers blood pressure compared to a placebo. Which type of hypothesis test should they use?

Accepted Answer

Two-sample t-test

Question 13

Which formula represents the joint probability of two independent events $$A$$ and $$B$$?

Accepted Answer

$$P(A \cap B) = P(A) \cdot P(B)$$

Question 14

Given the following probabilities: $$P(A) = 0.3$$, $$P(B) = 0.5$$, and $$P(A \cap B) = 0.15. $$Are events $$A$$ and $$B$$ independent?

Accepted Answer

Yes, because $$P(A \cap B) = P(A) \cdot P(B)$$

Question 15

A contingency table shows that 40 patients have diabetes and 60 do not. Of those with diabetes, 10 have high blood pressure. What is the conditional probability that a patient has high blood pressure given they have diabetes?

Accepted Answer

$$\frac{10}{40}$$

Question 16

Which of the following scenarios is most appropriate for a chi-squared test for independence?

Accepted Answer

Testing whether two categorical variables are associated.

Question 17

Which formula represents the sample correlation coefficient $$r$$ between variables $$X$$ and $$Y$$?

Accepted Answer

$$r = \frac{\sum (X_i - \bar{X})(Y_i - \bar{Y})}{(n-1)s_X s_Y}$$

Question 18

Which of the following is NOT something the correlation coefficient can tell us?

Accepted Answer

Whether the relationship is causal.

Question 19

A researcher wants to test whether there is no correlation between hours studied and exam scores. Which statistical test should they use?

Accepted Answer

t-test for lack of correlation

Applied Statistics Exam 2 Study Guide

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