T-test

Question 1

Multiple Choice:

1. In a population of adult men’s heights, random samples of size 50 have means ranging from 175 cm to 181 cm. Assuming the true population mean is 178 cm, what is the probability of drawing a sample with a mean of 176.5 cm or less? (Assume s = 7 cm)
   A) 0.1587
   B) 0.3413
   C) 0.5000
   D) 0.8413
2. The means of random samples (n = 50) from a population of adult men’s heights ranged from 175 cm to 181 cm. If the true population mean is assumed to be 178 cm, what is the probability of obtaining a sample mean of 176.5 cm or less? (Assume s = 7 cm)
   A) 0.1587
   B) 0.3413
   C) 0.5000
   D) 0.8413

Fill in the blanks:

1. In a population of adult men’s heights, if the true population mean is 178 cm and the standard deviation is 7 cm, the probability of drawing a sample with a mean of 176.5 cm or less can be calculated using the ___________ distribution.
2. The area under the curve to the left of 176.5 cm represents the probability of obtaining a sample mean ___________ or less.

Answer 1

Answers:

Multiple Choice:

1. A) 0.1587
2. A) 0.1587

Fill in the blanks:

1. normal
2. 176.5 cm

Question 2

Multiple Choice:

1. The degrees of freedom (df) is the number of:
   A) Parameters estimated
   B) Independent values used to calculate a parameter estimate
   C) Total values in the data set
   D) Samples collected
2. In statistics, degrees of freedom represent:
   A) The flexibility of making calculations
   B) The accuracy of parameter estimates
   C) The number of parameters estimated
   D) The size of the sample

Fill in the blanks:

1. Degrees of freedom (df) is calculated by subtracting the number of ___________ estimated from the total number of values used to estimate a parameter.
2. The accuracy of parameter estimates is influenced by the ___________ of the sample.

Answer 2

Answers:

Multiple Choice:

1. B) Independent values used to calculate a parameter estimate
2. B) The accuracy of parameter estimates

Fill in the blanks:

1. Parameters
2. Size

Question 3

Multiple Choice:

1. The t-distribution is similar to the normal distribution in terms of:
   A) Shape
   B) Skewness
   C) Tails
   D) Mean
2. Compared to the normal distribution, the t-distribution has:
   A) Lighter tails
   B) Symmetry
   C) Heavier tails
   D) Lower mean

Fill in the blanks:

1. The t-distribution is more prone to producing values that fall _________ from its mean compared to the normal distribution.
2. The t-distribution is characterized by its ___________ and bell-shaped nature.

Answer 3

Answers:

Multiple Choice:

1. A) Shape
2. C) Heavier tails

Fill in the blanks:

1. Far
2. Symmetric

Question 4

Analytical Questions:

1. What is the purpose of constructing a confidence interval for a single population?
2. In a one-sample t-test, what does the hypothesized/test population mean represent?
3. Why is the one-sample t-test considered the least common among the t-tests?

Subjective Question:
Explain the concept of a confidence interval for a single population mean in your own words.

Multiple Choice:
1. A confidence interval for a single population is used to estimate the true population:
A) Standard deviation
B) Variance
C) Proportion
D) Mean

2. In a one-sample t-test, the hypothesized/test population mean is:
   A) The mean of the sample
   B) The mean of the entire population
   C) The mean of a different population
   D) The mean of the sample and population combined

Fill in the Blanks:
1. A one-sample t-test is used to test for a difference between a population mean and a ________ population mean.
2. The least common among the t-tests is the ________.

Answer 4

Answers:

Analytical Questions:
1. The purpose of constructing a confidence interval for a single population is to estimate the true population mean.
2. In a one-sample t-test, the hypothesized/test population mean represents the mean of a different population or a value we want to compare our sample mean to.
3. The one-sample t-test is considered the least common among the t-tests because it is specific to situations where we want to compare a sample mean to a hypothesized/test population mean.

Subjective Question: The answer will vary based on the individual’s understanding and explanation of the concept.

Multiple Choice:
1. D) Mean
2. C) The mean of a different population

Fill in the Blanks:
1. Hypothesized/test
2. One-sample t-test

Question 5

Analytical Questions:

1. What is the purpose of conducting a one-sample t-test?
2. What are the assumptions related to the study design in a one-sample t-test?
3. What are the data assumptions for conducting a one-sample t-test?

Subjective Question:
Explain why the assumption of approximately normally distributed data is important in a one-sample t-test.

Multiple Choice:
1. In a one-sample t-test, the dependent variable should be:
A) Categorical
B) Discrete
C) Continuous
D) Binary

2. Which of the following is an assumption related to the study design in a one-sample t-test?
   A) Normal distribution of data
   B) No extreme outliers
   C) Independence of observations
   D) Single sample design

Fill in the Blanks:
1. One of the data assumptions for a one-sample t-test is that there are no ________.
2. The study design assumption for a one-sample t-test includes the ________ of observations.

Answer 5

Answers:

Analytical Questions:
1. The purpose of conducting a one-sample t-test is to compare the mean of a single sample to a test/hypothesized mean and determine if there is a significant difference.
2. The assumptions related to the study design in a one-sample t-test include having a single sample and a dependent variable that is continuous. Additionally, the data should be independent, meaning that observations are unrelated.
3. The data assumptions for conducting a one-sample t-test are that there should be no extreme outliers, and the data should be approximately normally distributed.

Subjective Question: The answer will vary based on the individual’s understanding and explanation of the importance of the assumption of approximately normally distributed data in a one-sample t-test.

Multiple Choice:
1. C) Continuous
2. D) Single sample design

Fill in the Blanks:
1. No extreme outliers
2. Independence of observations

Question 6

Frame multiple-choice, fill in the blanks, and analytical questions based on the given information:

Multiple Choice:
1. In a one-sample t-test, the null hypothesis (𝐻0) states that:
A) The means of two populations are equal
B) The means of two samples are equal
C) The mean of the sample is equal to the hypothesized/test mean
D) The variances of two populations are equal

2. Which of the following is the alternative hypothesis (𝐻𝐴) for a one-sample t-test when the study question aims to determine if the mean is different from the hypothesized/test mean?
   A) 𝐻𝐴: 𝜇 ≠ 𝜇0 (2-tailed)
   B) 𝐻𝐴: 𝜇 < 𝜇0 (1-tailed)
   C) 𝐻𝐴: 𝜇 > 𝜇0 (1-tailed)
   D) 𝐻𝐴: 𝜇 = 𝜇0

Answer 6

Fill in the Blanks:
1. In a one-sample t-test, the null hypothesis is always stated as 𝐻0: 𝜇 = 𝜇0, where 𝜇0 represents the hypothesized/test mean.
2. The alternative hypothesis for a one-sample t-test can be 𝐻𝐴: 𝜇 ≠ 𝜇0 (2-tailed), 𝐻𝐴: 𝜇 < 𝜇0 (1-tailed), or 𝐻𝐴: 𝜇 > 𝜇0 (1-tailed), depending on the study question.

Analytical Question:
Explain the difference between a two-tailed and one-tailed alternative hypothesis in a one-sample t-test and when each might be appropriate to use.

Note: The answers for the questions will depend on the context and understanding of the individual.

Question 7

The recommended daily intake (RDI) of calcium for women 40-50 years old is
1000 mg. A research group had reason to believe that women were not meeting
this requirement. To assess mean calcium intake (mg/day), the team recruited 85
participants and surveyed their calcium intake.
Sample statistics: 𝑥ҧ = 978.9306 and 𝑠 = 280.5664
(Note: retain as many decimal places as possible through calculations)
What is the true population mean? Calculate the 95% CIs for the mean

Answer 7

To calculate the true population mean and the 95% confidence interval (CI), we can use the sample mean and sample standard deviation along with the formula for the CI.

Given:
Sample mean (𝑥ҧ) = 978.9306
Sample standard deviation (𝑠) = 280.5664
Sample size (𝑛) = 85

To calculate the true population mean, we can use the sample mean:
True population mean (𝜇) = 𝑥ҧ = 978.9306

To calculate the 95% confidence interval for the mean, we need to consider the t-distribution since the sample size is less than 30.

Step 1: Find the critical value (t*) for a 95% confidence level with 𝑛-1 degrees of freedom.
The degrees of freedom (df) for a one-sample t-test is 𝑛-1.
df = 85 - 1 = 84
Using a t-table or a statistical software, we find that the critical value for a 95% confidence level and df = 84 is approximately 1.990.

Step 2: Calculate the standard error (SE) of the mean.
SE = 𝑠 / √𝑛
SE = 280.5664 / √85 ≈ 30.5157

Step 3: Calculate the margin of error (ME).
ME = t* × SE
ME = 1.990 × 30.5157 ≈ 60.7382

Step 4: Calculate the lower and upper bounds of the confidence interval.
Lower bound = 𝑥ҧ - ME
Lower bound = 978.9306 - 60.7382 ≈ 918.1924

Upper bound = 𝑥ҧ + ME
Upper bound = 978.9306 + 60.7382 ≈ 1039.6688

The 95% confidence interval for the true population mean is approximately 918.1924 to 1039.6688.

Therefore, the true population mean is estimated to be 978.9306 mg/day, with a 95% confidence interval of 918.1924 to 1039.6688 mg/day.
To find the test statistic and p-value, we can perform a one-sample t-test using the given sample statistics and the null hypothesis 𝐻0: 𝜇 = 𝜇0, where 𝜇0 is the hypothesized mean.

Given:
Sample mean (𝑥ҧ) = 978.9306
Sample standard deviation (𝑠) = 280.5664
Sample size (𝑛) = 85
Hypothesized mean (𝜇0) = 1000 (RDI of calcium for women 40-50 years old)

Step 1: Calculate the test statistic (t-score).
t = (𝑥ҧ - 𝜇0) / (𝑠 / √𝑛)
t = (978.9306 - 1000) / (280.5664 / √85) ≈ -1.6122

Step 2: Find the p-value associated with the test statistic.
Using a t-table or a statistical software, we find that the p-value for a t-score of -1.6122 with df = 84 is approximately 0.1108.

Step 3: Make a statistical decision and conclusion.
Since the p-value (0.1108) is greater than the significance level (e.g., α = 0.05), we fail to reject the null hypothesis. This means that there is not enough evidence to conclude that the mean calcium intake for women 40-50 years old is significantly different from the recommended daily intake of 1000 mg.

Therefore, based on the sample data, we do not have sufficient evidence to suggest that women in this age group are not meeting the recommended calcium requirement.

Question 8

Multiple Choice Questions:
1. The paired t-test is used to test for a mean difference between:
a) Two independent groups
b) Two related groups
c) Two different populations
d) Two unrelated variables

2. The paired t-test is appropriate when:
   a) The measurements in one group influence the measurements of the other
   b) The groups are independent of each other
   c) The populations being compared are similar in size
   d) The data are normally distributed

Fill in the Blanks:
1. The paired t-test is commonly used when comparing measurements taken at ____________ timepoints or when each participant undergoes ____________ interventions.
2. The paired t-test compares the ____________ mean difference between the two groups to determine if it is statistically significant.

(Note: The answers may vary depending on the context and specific wording of the questions)

Answer 8

Multiple Choice Questions:
1. b) Two related groups
2. a) The measurements in one group influence the measurements of the other

Fill in the Blanks:
1. different; both
2. mean

Question 9

Multiple Choice Questions:
1. The paired t-test is used to test for a mean difference between:
a) Two independent groups
b) Two related groups
c) Two different populations
d) Two unrelated variables

2. The paired t-test is appropriate when:
   a) The measurements in one group influence the measurements of the other
   b) The groups are independent of each other
   c) The populations being compared are similar in size
   d) The data are normally distributed

Fill in the Blanks:
1. The paired t-test is commonly used when comparing measurements taken at ____________ timepoints or when each participant undergoes ____________ interventions.
2. The paired t-test compares the ____________ mean difference between the two groups to determine if it is statistically significant.

(Note: The answers may vary depending on the context and specific wording of the questions)

Answer 9

Multiple Choice Questions:
1. b) Two related groups
2. a) The measurements in one group influence the measurements of the other

Fill in the Blanks:
1. different; both
2. mean

Question 10

Multiple Choice Questions:
1. What does 𝐻0 represent in the hypothesis statements for a paired t-test?
a) The null hypothesis
b) The alternative hypothesis
c) The difference between the paired observations
d) The mean of the dependent variable

2. Which of the following is an alternative hypothesis for a paired t-test?
   a) 𝐻𝐴: 𝜇𝐷 = 0
   b) 𝐻𝐴: 𝜇𝐷 < 0
   c) 𝐻𝐴: 𝜇𝐷 = 𝜇1 − 𝜇2
   d) 𝐻𝐴: 𝜇1 = 𝜇2

Fill in the Blanks:
1. In the paired t-test, 𝐻0 assumes that the mean difference (𝜇𝐷) is _______.
2. The alternative hypothesis for a paired t-test can be 𝜇𝐷 ≠ 0, 𝜇𝐷 > 0, or 𝜇𝐷 < 0, indicating a _______ in the mean difference.

Answer 10

Answers:
Multiple Choice:
1. a) The null hypothesis
2. b) 𝐻𝐴: 𝜇𝐷 < 0

Fill in the Blanks:
1. 0
2. change

Question 11

Multiple Choice:
1. The independent t-test is used to test for a mean difference between which type of groups?
a) Dependent groups
b) Related groups
c) Independent groups
d) Categorical groups

2. In which of the following contexts is the independent t-test commonly used?
   a) Two timepoints for the same observations
   b) Each participant undergoing both interventions
   c) Two different observed states (disease vs healthy)
   d) Two groups receiving the same treatment

Fill in the Blanks:
1. The independent t-test is appropriate when the measurements in one group ________ influence the measurement of the other.

2. The independent t-test compares the means of two ________ groups.
3. In the independent t-test, the null hypothesis states that there is ________ mean difference between the two groups.
4. The alternative hypothesis in the independent t-test can be one-sided, indicating a mean difference in a specific ________, or two-sided, indicating a mean difference that is not equal to ________.

Critical Question:
Explain the difference between dependent groups and independent groups in the context of the independent t-test. Provide an example for each type of group.

Answer 11

Answer: c) Independent groups

Answer: c) Two different observed states (disease vs healthy)
Answer: do not
Answer: independent
Answer: no
Answer: direction, zero

Critical Question Answer:
In the context of the independent t-test, dependent groups refer to cases where the measurements in one group are influenced by or related to the measurements in the other group. This means that the observations within each group are not independent. An example of dependent groups could be measuring the effectiveness of a new drug by comparing the pre-treatment and post-treatment values of the same individuals.

On the other hand, independent groups are those where the measurements in one group are not influenced by the measurements in the other group. The observations within each group are independent of each other. For example, comparing the test scores of students who attended a tutoring program versus those who did not attend would involve independent groups.

The distinction between dependent and independent groups is important because the independent t-test assumes independence between the groups. Violation of this assumption can lead to incorrect results.