Master Continuous Probability Distributions in Statistics: Quizzes and Practice

Question 1

Which characteristic uniquely defines the Normal distribution?

Accepted Answer

Symmetry around the mean

Answer

Exclusively defined for positive values

Answer

Equal variance and mean

Answer

Discrete nature of distribution

Question 2

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

Accepted Answer

Mean is equal to the median

Answer

Symmetrical about the mean

Answer

Bell-shaped

Answer

Total area under the curve is 1

Question 3

Which of the following is the distribution of the sample mean when the population is normally distributed?

Accepted Answer

Normal

Answer

t

Answer

Chi-square

Answer

Exponential

Question 4

The chi-square test of independence is used to determine whether:

Accepted Answer

Two categorical variables are independent

Answer

A categorical variable follows a specific distribution

Answer

A continuous variable is normally distributed

Answer

A sample mean is significantly different from a population mean

Question 5

Which of the following is an example of an application of the t-distribution?

Accepted Answer

Testing the significance of a sample mean

Answer

Estimating a population proportion

Answer

Analyzing categorical data

Answer

Modeling continuous data

Question 6

Using a t-distribution table, what is the probability of obtaining a t-score between -2.576 and 2.576 for a t-distribution with 10 degrees of freedom?

Accepted Answer

0.05

Answer

0.01

Answer

0.025

Answer

0.95

Question 7

How do the population mean and sample mean differ in the context of the Normal distribution?

Accepted Answer

The population mean is a fixed value, while the sample mean is an estimate based on a sample from the population.

Answer

The sample mean is a fixed value, while the population mean is an estimate based on the entire population.

Answer

The population mean and sample mean are always equal.

Answer

There is no difference between the population mean and sample mean.

Question 8

The standard Normal distribution is the distribution of all _____.

Accepted Answer

Normal distributions

Answer

t-values

Answer

chi-square values

Answer

sample means

Question 9

The t-distribution is used for testing hypotheses about _____.

Accepted Answer

Population means when the sample size is small

Answer

Population means when the sample size is large

Answer

Population proportions

Answer

Population variances

Question 10

The chi-square distribution is used for testing hypotheses about _____.

Accepted Answer

Population proportions

Answer

Population means

Answer

Population variances

Answer

Correlation coefficients

Question 11

The sample size required to use the Normal distribution for inference can be determined by _____.

Accepted Answer

The Central Limit Theorem

Answer

The t-distribution

Answer

The chi-square distribution

Answer

The F-distribution

Question 12

The chi-square goodness-of-fit test is used to test whether _____.

Accepted Answer

The observed frequencies of a categorical variable match the expected frequencies

Answer

The mean of a sample is equal to the population mean

Answer

The variance of a sample is equal to the population variance

Answer

The correlation between two variables is zero

Question 13

Which of the following distributions is symmetric?

Accepted Answer

Normal distribution

Answer

t-distribution

Answer

Chi-square distribution

Answer

F-distribution

Question 14

Which of the following is NOT a characteristic of the Normal distribution?

Accepted Answer

Constant mean and variance

Answer

Symmetrical

Answer

Unimodal

Answer

Bell-shaped

Question 15

The t-distribution is used for making inferences about:

Accepted Answer

Population mean when the sample size is small and the population standard deviation is unknown

Answer

Population mean when the sample size is large

Answer

Population proportion

Answer

Population variance

Question 16

Which of the following is a key difference between the t-distribution and the Normal distribution?

Accepted Answer

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

Answer

The t-distribution is skewed, while the Normal distribution is symmetric.

Answer

The t-distribution has a lower mean than the Normal distribution.

Answer

The t-distribution has a higher variance than the Normal distribution.

Question 17

The degrees of freedom for a chi-square distribution with k categories is:

Accepted Answer

k - 1

Answer

k

Answer

k + 1

Answer

k / 2

Question 18

If the chi-square statistic for a goodness-of-fit test is large, what is the most likely conclusion?

Accepted Answer

The observed frequencies are significantly different from the expected frequencies.

Answer

The observed frequencies are not significantly different from the expected frequencies.

Answer

The sample size is too small.

Answer

The data is not normally distributed.

Question 19

Which of the following accurately describes a key characteristic of the Normal distribution?

Accepted Answer

Symmetrical bell-shaped curve

Answer

Mean, median, and mode that are all equivalent

Answer

Total area beneath the curve that is infinite

Answer

Standard deviation that is always negative

Question 20

Which of the following scenarios best exemplifies a common application of the chi-square distribution?

Accepted Answer

Assessing the independence of categorical variables

Answer

Determining how well a model fits observed data

Answer

Estimating a population mean

Answer

Testing a sample variance

Question 21

As the degrees of freedom approach infinity, how does the t-distribution compare to the Normal distribution?

Accepted Answer

It becomes identical to the Normal distribution

Answer

It becomes narrower than the Normal distribution

Answer

It becomes wider than the Normal distribution

Answer

It becomes negatively skewed

Question 22

What is the probability density function (PDF) of the Normal distribution?

Accepted Answer

f(x) = (1/√(2πσ^2)) * e^(-(x-μ)^2/(2σ^2))

Answer

f(x) = kx^2

Answer

f(x) = e^(-x^2)

Answer

f(x) = 1/(2πσ^2)

Question 23

When is it appropriate to use the t-distribution for statistical inference?

Accepted Answer

When the population standard deviation is unknown and the sample size is small

Answer

When the population standard deviation is known

Answer

When the population standard deviation is unknown and the sample size is large

Answer

When the population standard deviation is always unknown

Question 24

What is a potential limitation of the chi-square distribution?

Accepted Answer

Assumes expected frequencies are not too small

Answer

Only applicable to categorical variables

Answer

Cannot be used for hypothesis testing

Answer

Assumes a Normal population distribution

Question 25

The t-distribution has heavier tails than the Normal distribution when the sample size is:

Accepted Answer

Small

Answer

Large

Answer

Equal

Answer

Unknown

Question 26

For a random sample from a population with mean μ and variance σ^2, the standard error of the sample mean is:

Accepted Answer

σ/√n

Answer

μ/√n

Answer

σ^2/√n

Answer

μ√n

Question 27

Which of the following is NOT an application of the t-distribution?

Accepted Answer

ANOVA

Answer

Hypothesis testing for a population mean

Answer

Confidence interval estimation for a population mean

Answer

Comparing the means of two independent populations

Question 28

The chi-square goodness-of-fit test is used to:

Accepted Answer

Test whether a sample comes from a specified distribution

Answer

Compare the means of two populations

Answer

Estimate the variance of a population

Answer

Test for independence in a contingency table

Question 29

In a hypothesis test for a population mean, the alternative hypothesis is:

Accepted Answer

The population mean is not equal to a specific value

Answer

The population mean is equal to a specific value

Answer

The population mean is greater than a specific value

Answer

The population mean is less than a specific value

Question 30

A researcher wants to test the effectiveness of a new drug. They collect data from a sample of 25 patients and want to determine if the average change in blood pressure is significantly different from zero. What distribution would be most appropriate to use for this hypothesis test?

Accepted Answer

t-distribution

Answer

Normal distribution

Answer

F-distribution

Answer

Chi-square distribution

Question 31

Which of the following is NOT a property of the chi-square distribution?

Accepted Answer

It is symmetric about its mean.

Answer

Its mean is equal to its degrees of freedom.

Answer

It is used in hypothesis tests for categorical data.

Answer

It is positively skewed.

Question 32

A company wants to compare the effectiveness of two different marketing campaigns. They collect data on the number of sales generated by each campaign and want to test if there is a significant difference in the average sales. What statistical test would be most appropriate to use?

Accepted Answer

Two-sample t-test

Answer

Chi-square test

Answer

One-sample t-test

Answer

Paired t-test

Question 33

The standard normal distribution has a mean of _____ and a standard deviation of _____.

Accepted Answer

0, 1

Answer

0, 0

Answer

1, 1

Answer

1, 0

Question 34

A researcher is studying the effects of a new fertilizer on plant growth. They randomly assign plants to two groups: one group receives the new fertilizer, and the other group receives a control fertilizer. After a period of time, they measure the height of each plant. What statistical test would be most appropriate to analyze the data?

Accepted Answer

Two-sample t-test

Answer

Chi-square test

Answer

One-sample t-test

Answer

Paired t-test

Question 35

A company wants to estimate the average lifespan of its products. They collect data from a random sample of products and find that the sample mean lifespan is 5 years. The company wants to construct a 95% confidence interval for the true average lifespan. What distribution would be used to determine the critical value for the confidence interval?

Accepted Answer

t-distribution

Answer

F-distribution

Answer

Chi-square distribution

Answer

Normal distribution

Question 36

What is the relationship between the degrees of freedom and the shape of the t-distribution?

Accepted Answer

As the degrees of freedom increase, the t-distribution becomes more similar to the standard normal distribution.

Answer

As the degrees of freedom increase, the t-distribution becomes more skewed.

Question 37

Which of the following is NOT a characteristic of the t-distribution?

Accepted Answer

It is always symmetric about zero.

Answer

It has heavier tails than the Normal distribution.

Answer

It is bell-shaped.

Answer

Its shape depends on the degrees of freedom.

Question 38

A researcher wants to test the hypothesis that the average weight of a certain type of fruit is 100 grams.  She collects a sample of 20 fruits and finds the sample mean weight to be 98 grams.  The population standard deviation is known to be 5 grams. What is the appropriate test statistic to use for this hypothesis test?

Accepted Answer

z-statistic

Answer

t-statistic

Answer

F-statistic

Answer

chi-square statistic

Question 39

A random sample of 100 observations is taken from a population with a standard deviation of 15. The sample mean is found to be 50. What is the standard error of the mean?

Accepted Answer

1.5

Answer

0.15

Answer

15

Question 40

The area under the curve of a probability distribution always represents:

Accepted Answer

The probability of an event occurring within a specified range.

Answer

The mean of the distribution.

Answer

The standard deviation of the distribution.

Question 41

Which of the following is TRUE about the Normal distribution?

Accepted Answer

It is completely defined by its mean and standard deviation.

Answer

It has a limited number of possible values.

Answer

It is always skewed to the right.

Question 42

A researcher is interested in comparing the average lifespan of two different breeds of dogs.  She collects data on a sample of 20 dogs from each breed.  What statistical test should she use to compare the mean lifespans of the two breeds?

Accepted Answer

Two-sample t-test.

Answer

One-sample t-test.

Answer

Z-test.

Answer

Chi-square test.

Question 43

A researcher wants to determine if there is a relationship between the number of hours students study per week and their final exam scores.  What statistical test should she use to analyze the data?

Accepted Answer

Correlation.

Answer

Chi-square test.

Answer

Regression.

Answer

One-way ANOVA.