Solved on Nov 24, 2023

Test whether there is a significant difference between the IQs of spouses, given IQ data for 9 married couples. Assume the paired difference distribution is approximately normal. Use a 0.10 significance level.

STEP 1

Assumptions1. The IQ scores of the9 couples are given. . We are testing the claim that there is a significant difference between the IQs of spouses.
3. We are using a0.10 level of significance.
4. The population distribution of the paired differences is approximately normal.
5. Spouse1 group is Population1 and Spouse group is Population.

STEP 2

First, we need to calculate the difference in IQ scores for each couple. This is done by subtracting the IQ of Spouse2 from the IQ of Spouse1 for each couple.
$= IQ_{Spouse1} - IQ_{Spouse2}$

STEP 3

Calculate the differences for each couple.
$= [118-114,117-112,124-127,113-109,106-102,128-129,120-124,114-110,119-116]$

STEP 4

Calculate the numerical differences.
$= [4,, -3,4,4, -1, -4,4,3]$

STEP 5

Now, we need to calculate the mean of these differences. This is done by summing all the differences and dividing by the number of couples (9).
$\bar{} = \frac{\sum D}{n}$

STEP 6

Calculate the mean difference.
$\bar{} = \frac{4 +5 -3 +4 +4 -1 -4 +4 +3}{9}$

STEP 7

Calculate the numerical mean difference.
$\bar{} = \frac{16}{9} \approx1.778$

STEP 8

Next, we need to calculate the standard deviation of the differences. The standard deviation is calculated by taking the square root of the variance. The variance is the average of the squared differences from the mean.
$s = \sqrt{\frac{\sum ( - \bar{})^2}{n-1}}$

STEP 9

Calculate the standard deviation.
$s = \sqrt{\frac{(4-.778)^2 + (5-.778)^2 + (-3-.778)^2 + (4-.778)^2 + (4-.778)^2 + (--.778)^2 + (-4-.778)^2 + (4-.778)^2 + (3-.778)^2}{9-}}$

STEP 10

Calculate the numerical standard deviation.
$s \approx3.202$

STEP 11

Now, we can calculate the test statistic for a paired t-test. The test statistic is calculated by dividing the mean difference by the standard error. The standard error is the standard deviation divided by the square root of the number of couples.
$t = \frac{\bar{}}{s/\sqrt{n}}$

STEP 12

Calculate the test statistic.
$t = \frac{.778}{.202/\sqrt{9}}$

STEP 13

Calculate the numerical test statistic.
$t \approx.667$ The test statistic is approximately.667.

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Test whether there is a significant difference between the IQs of spouses, given IQ data for 9 married couples. Assume the paired difference distribution is approximately normal. Use a 0.10 significance level.

STEP 1

STEP 2

First, we need to calculate the difference in IQ scores for each couple. This is done by subtracting the IQ of Spouse2 from the IQ of Spouse1 for each couple.
$= IQ_{Spouse1} - IQ_{Spouse2}$

STEP 3

Calculate the differences for each couple.
$= [118-114,117-112,124-127,113-109,106-102,128-129,120-124,114-110,119-116]$

STEP 4

Calculate the numerical differences.
$= [4,, -3,4,4, -1, -4,4,3]$

STEP 5

Now, we need to calculate the mean of these differences. This is done by summing all the differences and dividing by the number of couples (9).
$\bar{} = \frac{\sum D}{n}$

STEP 6

Calculate the mean difference.
$\bar{} = \frac{4 +5 -3 +4 +4 -1 -4 +4 +3}{9}$

STEP 7

Calculate the numerical mean difference.
$\bar{} = \frac{16}{9} \approx1.778$

STEP 8

Next, we need to calculate the standard deviation of the differences. The standard deviation is calculated by taking the square root of the variance. The variance is the average of the squared differences from the mean.
$s = \sqrt{\frac{\sum ( - \bar{})^2}{n-1}}$

STEP 9

Calculate the standard deviation.
$s = \sqrt{\frac{(4-.778)^2 + (5-.778)^2 + (-3-.778)^2 + (4-.778)^2 + (4-.778)^2 + (--.778)^2 + (-4-.778)^2 + (4-.778)^2 + (3-.778)^2}{9-}}$

STEP 10

Calculate the numerical standard deviation.
$s \approx3.202$

STEP 11

Now, we can calculate the test statistic for a paired t-test. The test statistic is calculated by dividing the mean difference by the standard error. The standard error is the standard deviation divided by the square root of the number of couples.
$t = \frac{\bar{}}{s/\sqrt{n}}$

STEP 12

Calculate the test statistic.
$t = \frac{.778}{.202/\sqrt{9}}$

STEP 13

Calculate the numerical test statistic.
$t \approx.667$ The test statistic is approximately.667.

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Test whether there is a significant difference between the IQs of spouses, given IQ data for 9 married couples. Assume the paired difference distribution is approximately normal. Use a 0.10 significance level.

STEP 1

STEP 2

First, we need to calculate the difference in IQ scores for each couple. This is done by subtracting the IQ of Spouse2 from the IQ of Spouse1 for each couple.=IQSpouse1−IQSpouse2 = IQ_{Spouse1} - IQ_{Spouse2}=IQSpouse1​−IQSpouse2​

STEP 3

STEP 4

Calculate the numerical differences.=[4,,−3,4,4,−1,−4,4,3] = [4,, -3,4,4, -1, -4,4,3]=[4,,−3,4,4,−1,−4,4,3]

STEP 5

Now, we need to calculate the mean of these differences. This is done by summing all the differences and dividing by the number of couples (9).ˉ=∑Dn\bar{} = \frac{\sum D}{n}ˉ=n∑D​

STEP 6

Calculate the mean difference.ˉ=4+5−3+4+4−1−4+4+39\bar{} = \frac{4 +5 -3 +4 +4 -1 -4 +4 +3}{9}ˉ=94+5−3+4+4−1−4+4+3​

STEP 7

Calculate the numerical mean difference.ˉ=169≈1.778\bar{} = \frac{16}{9} \approx1.778ˉ=916​≈1.778

STEP 8

STEP 9

STEP 10

Calculate the numerical standard deviation.s≈3.202s \approx3.202s≈3.202

STEP 11

Now, we can calculate the test statistic for a paired t-test. The test statistic is calculated by dividing the mean difference by the standard error. The standard error is the standard deviation divided by the square root of the number of couples.t=ˉs/nt = \frac{\bar{}}{s/\sqrt{n}}t=s/n​ˉ​

STEP 12

Calculate the test statistic.t=.778.202/9t = \frac{.778}{.202/\sqrt{9}}t=.202/9​.778​

STEP 13

Calculate the numerical test statistic.t≈.667t \approx.667t≈.667The test statistic is approximately.667.

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

First, we need to calculate the difference in IQ scores for each couple. This is done by subtracting the IQ of Spouse2 from the IQ of Spouse1 for each couple.
$= IQ_{Spouse1} - IQ_{Spouse2}$

Calculate the numerical differences.
$= [4,, -3,4,4, -1, -4,4,3]$

Now, we need to calculate the mean of these differences. This is done by summing all the differences and dividing by the number of couples (9).
$\bar{} = \frac{\sum D}{n}$

Calculate the mean difference.
$\bar{} = \frac{4 +5 -3 +4 +4 -1 -4 +4 +3}{9}$

Calculate the numerical mean difference.
$\bar{} = \frac{16}{9} \approx1.778$

Calculate the numerical standard deviation.
$s \approx3.202$

Now, we can calculate the test statistic for a paired t-test. The test statistic is calculated by dividing the mean difference by the standard error. The standard error is the standard deviation divided by the square root of the number of couples.
$t = \frac{\bar{}}{s/\sqrt{n}}$

Calculate the test statistic.
$t = \frac{.778}{.202/\sqrt{9}}$

Calculate the numerical test statistic.
$t \approx.667$ The test statistic is approximately.667.