StatTutor Lesson - Matched Pairs t Procedures

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Matched Pairs t Procedures

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Question 1

Question 1.

True or false: The most common use of a one-sample t procedure is to analyze matched pairs data.

True

False

Correct. Analyzing matched pairs data is the most common use of a one-sample t test.

Incorrect. Analyzing matched pairs data is the most common use of a one-sample t test.

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Question 2

118

Question 2.

When performing an experiment with paired individuals, how is randomization implemented?

Randomly assigning the paired individuals to the two treatments within each pair

Randomly allocating pairs of individuals to treatments

Randomizing the order in which each individual receives the two treatments

Randomly selecting the paired individuals

Correct. When individuals are paired in an experiment, they need to be randomly allocated to the two treatments.

Incorrect. When individuals are paired in an experiment, they need to be randomly allocated to the two treatments.

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Question 3

155

Question 3.

What do we compute for each pair of individuals?

The product of their two measurements

The difference between their two measurements

The sum of their two measurements

Correct. For each pair, we compute the difference between their two measurements.

Incorrect. For each pair, we compute the difference between their two measurements.

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Question 4

174

Question 4.

True or false: To analyze matched pairs data we always compute the mean and standard deviation of the differences.

True

False

Correct. This is a true statement.

Incorrect. This is a true statement.

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Question 5

213

Question 5.

When performing an experiment with two measurements on each individual, how is randomization implemented?

Randomizing the order in which each individual receives the two treatments

Randomly selecting the individuals for the study

Randomly assigning the paired individuals to the two treatments within each pair

Randomly allocating individuals to the two treatments

Correct. When two measurements are taken on each individual in an experiment, the order in which the treatments are applied needs to be randomized.

Incorrect. When two measurements are taken on each individual in an experiment, the order in which the treatments are applied needs to be randomized.

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Questions 6-7

228

Question 6.

In what way is a matched pairs analysis with two measurements on each individual considered a one-sample t procedure?

Because one-sample of differences is analyzed, not the actual measurements.

Because the two measurements are added to get one measurement for each pair of individuals.

Correct. The one sample of differences from a matched pairs study is analyzed with a one-sample t procedure.

Incorrect. The one sample of differences from a matched pairs study is analyzed with a one-sample t procedure.

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Question 8

297

Question 8.

Where performing a study when pre-treatment and post-treatment measurements are taken on each individual, how is randomization implemented?

Randomly assigning the paired individuals to the two treatments within each pair

Randomly allocating individuals to the two treatments

Randomly selecting the individuals for the study

Randomizing the order in which each individual receives the two treatments

Correct. When pre- and post- treatment measurements are taken on each individual, the individuals need to be randomly selected for the study.

Incorrect. When pre- and post- treatment measurements are taken on each individual, the individuals need to be randomly selected for the study.

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Question 9

487

Question 9.

True or false: If there is no difference between two measurements, then the difference between the two measurements equals zero.

False

True

Correct. No difference equates to a difference of zero.

Incorrect. No difference equates to a difference of zero.

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Question 10

569

Question 10.

When analyzing matched pairs data, what plot(s) should be examined for outliers?

Two plots, one for each set of measurements

Only the plot of the sample of differences

Correct. We analyze the differences for each pair when we analyze data for a matched pairs study. This includes plotting the differences and examining the plot for outliers.

Incorrect. We analyze the differences for each pair when we analyze data for a matched pairs study. This includes plotting the differences and examining the plot for outliers.

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Question 11

644

Question 11.

What value should be used for µ₀ in the test statistic $t = \frac{\overline{x} - \mu _{o}}{\frac{s}{ \sqrt{n}}}$ for a matched pairs t test?

Zero

One

Depends on the research question

Correct. If there is no difference, then the mean of the differences should be zero.

Incorrect. If there is no difference, then the mean of the differences should be zero.

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Question 12

714

Question 12.

If we want to determine whether the average sweetness has decreased during storage, what alternative hypothesis do we want to test? Note that the differences are computed as d_i = fresh minus stored.

H_a: µ_d > 0

H_a: µ_d < 0

H_a: µ_d ≠ 0

H_a: µ_d = 0

Correct. If sweetness decreased, then fresh sweetness will be larger than stored sweetness and the mean difference will be greater than zero or H_a: µ_d > 0.

Incorrect. If sweetness decreased, then fresh sweetness will be larger than stored sweetness and the mean difference will be greater than zero or H_a: µ_d > 0.

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Question 13

750

Question 13.

In context, what is µ_d for this study?

The mean difference in cola sweetness (fresh sweetness minus stored sweetness)

The mean sweetness of fresh cola

The mean sweetness of stored cola

Correct. µ_d represents the mean difference. Sweetness is measured and the difference equals fresh minus stored. So µ_d represents the mean difference in sweetness (fresh minus stored).

Incorrect. µ_d represents the mean difference. Sweetness is measured and the difference equals fresh minus stored. So µ_d represents the mean difference in sweetness (fresh minus stored).

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Question 14

793

Question 14.

How is randomization implemented in this experiment?

Randomly assigning whether a bottle will be tasted fresh or stored with the other bottle receiving the other treatment

Randomly allocating batches to be either tasted fresh or tasted after being stored

Randomizing the order in which each bottle gets tasted as fresh and tasted as stored

Randomly selecting the batches to be studied

Correct. Because the bottles are paired by belonging to the same batch, they need to be randomly allocated to the two treatments of being tasted fresh or tasted after being stored.

Incorrect. Because the bottles are paired by belonging to the same batch, they need to be randomly allocated to the two treatments of being tasted fresh or tasted after being stored.

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Question 15

819

Question 15.

True or false: Two of these differences are wrong because all differences should be positive.

True

False

Correct. Differences can be positive, negative or zero. These two differences are not wrong.

Incorrect. Differences can be positive, negative or zero. These two differences are not wrong.

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Question 16

843

Question 16.

What is s_d = 0.909?

The standard deviation of the initial cola sweetness for this sample of ten batches

The standard deviation of the sample of differences (fresh sweetness minus stored sweetness)

The standard deviation of the cola sweetness after being stored for this sample of ten batches

Correct. s_d = 0.909 is the standard deviation of the differences where each difference is computed as fresh sweetness minus stored sweetness.

Incorrect. s_d = 0.909 is the standard deviation of the differences where each difference is computed as fresh sweetness minus stored sweetness.

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Question 17

874

Question 17.

Are there any outliers in the plot of the ten differences?

Yes

Correct. There are no data points that cause us to say, “Oh, wow!!! That’s an outlier!"

Incorrect. There are no data points that cause us to say, “Oh, wow!!! That’s an outlier!"

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Question 18

900

Question 18.

What are the degrees of freedom associated with the test statistic $t = \frac{\overline{x} - \mu _{o}}{\frac{s}{ \sqrt{n}} }$ ?

Zero

Correct. df = number of differences minus one = 10 – 1 = 9.

Incorrect. df = number of differences minus one = 10 – 1 = 9.

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Questions 19-20

916

Question 19.

Since we are testing H_a: µ_d > 0, is this test one-sided or two-sided?

Two-sided

One-sided

Correct. With “>” in H_a, the test in one-sided.

Incorrect. With “>” in H_a, the test in one-sided.

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Question 21

935

Question 21.

Because there is insufficient evidence to say the cola lost sweetness after one month of storage, can we say for sure that the cola did NOT lose sweetness on average?

Yes

Correct. Even though we failed to reject H₀, we do NOT have evidence to say that H₀: µ_d = 0 is correct. Perhaps the sample size of 10 batches is too small to detect significance and we need a larger sample size.

Incorrect. Even though we failed to reject H₀, we do NOT have evidence to say that H₀: µ_d = 0 is correct. Perhaps the sample size of 10 batches is too small to detect significance and we need a larger sample size.

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Question 22

981

Question 22.

Why is this study matched pairs?

Rats are paired.

Two measurements were taken on each rat.

Pre- and post-treatment measurements were taken on each rat.

Correct. Since two incisions were made on the back of each rat and each incision was repaired with a different treatment, this is a matched pairs study.

Incorrect. Since two incisions were made on the back of each rat and each incision was repaired with a different treatment, this is a matched pairs study.

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Question 23

1035

Question 23.

For P-value = 0.0467 and α = 0.05, are these results statistically significant?

Yes

Correct. P-value = 0.0467 is smaller than α = 0.05, so we reject H₀ and declare the results statistically significant.

Incorrect. P-value = 0.0467 is smaller than α = 0.05, so we reject H₀ and declare the results statistically significant.

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Question 24

1074

Question 24.

Differences for this matched pairs test were computed as difference = taped strength minus sutured strength. The mean difference for the sample of 10 rats is = 144.7. Which was stronger: taped or sutured?

Taped

Sutured

Correct. Since the difference is positive, taped strength is greater than sutured strength.

Incorrect. Since the difference is positive, taped strength is greater than sutured strength.

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Question 25

1132

Question 25.

What does the interval $\overline{x}_d \pm t^{*} \frac{ S_{d} }{ \sqrt{n} }$ estimate?

The mean of the differences in a sample of differences for a matched pairs set of data.

The mean of the population of all possible differences

Correct. µ_d is the mean difference or the mean of the population of all possible differences.

Incorrect. µ_d is the mean difference or the mean of the population of all possible differences.

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Question 26

1176

Question 26.

In the context of this example, what is the parameter needing to be estimated with a confidence interval?

The mean time for all right-handed students to turn a knob a specified distance in the clockwise direction

The mean difference in time required for right-handed students to turn a counterclockwise knob versus a clockwise knob

The mean time for all right-handed students to turn a knob a specified distance in the counterclockwise direction

Correct. The parameter is the mean difference in time for all right-handed students where the difference is computed as difference = time to turn counterclockwise knob minus time to turn a clockwise knob.

Incorrect. The parameter is the mean difference in time for all right-handed students where the difference is computed as difference = time to turn counterclockwise knob minus time to turn a clockwise knob.

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Question 27

1236

Question 27.

What values are plotted in the dotplot?

Times to turn a clockwise knob

Times to turn a counterclockwise knob

Differences in time (time to turn a counterclockwise knob minus time to turn a clockwise knob)

Correct. When analyzing matched pairs data, we always plot the differences and check for outliers in the differences.

Incorrect. When analyzing matched pairs data, we always plot the differences and check for outliers in the differences.

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Question 28

1292

Question 28.

On the basis of the interval estimate for µ_d, namely (3.87, 22.77), can we conclude that µ_d differs from zero?

Yes

Correct. Since the interval for µ_d, namely (3.87, 22.77), does not contain the value of zero, we can conclude that µ_d is not zero.

Incorrect. Since the interval for µ_d, namely (3.87, 22.77), does not contain the value of zero, we can conclude that µ_d is not zero.

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Question 29

1306

Question 29.

Using the interval for µ_d, namely (3.87, 22.77) where difference = time to turn a counterclockwise knob minus time to turn a clockwise knob, which takes right-handed students longer to turn on average: counterclockwise knobs or clockwise knobs?

Clockwise knobs

Counterclockwise knobs

Correct. Since difference = time to turn a counterclockwise knob minus time to turn a clockwise knob and the entire interval is positive, the average time to turn a counterclockwise knob is greater than the average time to turn a clockwise knob.

Incorrect. Since difference = time to turn a counterclockwise knob minus time to turn a clockwise knob and the entire interval is positive, the average time to turn a counterclockwise knob is greater than the average time to turn a clockwise knob.

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StatTutor Lesson - Matched Pairs t Procedures

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