A psychologist believes that a lack of sleep before students take the SAT will lead to them getting lower Math SAT scores. It is known that the mean Math SAT score is μ = 500 with a standard deviation of σ = 100. To test this belief, the psychologist recruited n=8 students who were scheduled to take the SAT. The night before the SAT, the researcher only let the n=8 students sleep for 2 hours. The mean Math SAT score for the n=8 students was M2hrSleep = 435. Perform a t-test by answering the questions below. Use an alpha-level of α=.05. 1. What is the research hypothesis, H1? 2. What is the null hypothesis, H0? 3. What is the degrees of freedom for this problem, df? 4. What is the critical value of t, tcrit? (Be sure to use a + or a - sign for your answer.) 5. What is the mean of the distribution of sample means, μM? 6. What is the standard deviation of the distribution of sample means, σM? 7. What is the z-score of the sample mean, M2hrSleep? 7. Based on the relation between the critical value of t and the t-score of the sample mean, can you reject the null hypothesis? 8. Based on whether or not you can reject the null hypothesis, can you accept the research hypothesis? Show your work for all problems.

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Description: A psychologist believes that a lack of sleep before students take the SAT will lead to them getting lower Math SAT scores. It is known that the mean Math SAT score is μ = 500 with a standard deviation of σ = 100. To test this belief, the psychologi

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018

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ISBN:9780079039897

Author:Carter

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Chapter10: Statistics

Section10.4: Distributions Of Data

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A psychologist believes that a lack of sleep before students take the SAT will lead to them getting lower Math SAT scores. It is known that the mean Math SAT score is μ = 500 with a standard deviation of σ = 100. To test this belief, the psychologist recruited n=8 students who were scheduled to take the SAT. The night before the SAT, the researcher only let the n=8 students sleep for 2 hours. The mean Math SAT score for the n=8 students was M_2hrSleep = 435.

Perform a t-test by answering the questions below. Use an alpha-level of α=.05.

1. What is the research hypothesis, H₁?

2. What is the null hypothesis, H₀?

3. What is the degrees of freedom for this problem, df?

4. What is the critical value of t, t_crit? (Be sure to use a + or a - sign for your answer.)

5. What is the mean of the distribution of sample means, μ_M?

6. What is the standard deviation of the distribution of sample means, σ_M?

7. What is the z-score of the sample mean, M_2hrSleep?

7. Based on the relation between the critical value of t and the t-score of the sample mean, can you reject the null hypothesis?

8. Based on whether or not you can reject the null hypothesis, can you accept the research hypothesis?

Show your work for all problems.

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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