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# Importance of Pooled Variance

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1. What is pooled variance and why is it important?

2. Explain what interval data is and give an example:

3. Write the formula for a problem that has 2 sample populations greater than 30 and the standard deviations are known and equal:

4. Write the formula for pooled variance.

5. Please analyze the following data:

A school district wants to determine if the girls are equal to boys in test scores. They took a random sample of 22 8th grade girls and 24 8th grade boys. The district looked at recent CAT scores and found the mean score for girls to be xbar1 = 25 and the mean score for boys to be xbar2 = 26. The standard deviation for the girls is s1 = 2.2 and for the boys the standard deviation is s2 = 3.4. Is the school district correct in assuming the girls are equal in performance to boys. They are 95% sure their assumption is correct.

6. Please analyze the following data

A candy company wants to identify whether or not the size of its' candy bars are the same length from one day to another. On day 1 they sample 52 candy bars with an average mean of 5.4 inches and on day 2 they sample 52 bars again with and average mean of 5.6 inches. Both days have sample standard deviations of 3.4. Is the average mean of size different from day 1 to day 2?

7. A sample of 40 observations is selected from one population. The sample mean is 102 and the sample standard deviation is 5. A sample of 50 observations is selected from a second population. The sample mean is 99 and the sample standard deviation is 6. Conduct the following test using 95% level of confidence.
A. Is it a 1 or 2 tail test
B. Compute the statistical calculation
C. What is your decision regarding H1

8. A sample of 65 observations is selected from one population. The sample mean is 2.67 and the sample standard deviation is .75. A sample of 50 observations is selected from a second population. The sample mean is 2.59 and the sample standard deviation is .66. Conduct the following test using 95% level of confidence.
a. Is it a 1 or 2 tail tes
b. Compute the statistical calculation
c. What is your decision regarding H1

9. The Gibbs Baby Food Company wishes to compare the weight gain of infants using their brand versus their competitor's brand. A sample of 40 babies using the Gibbs products revealed a mean weight gain of 7.6 pounds in the first 3 months after birth. The standard deviation of this sample is 2.3 pounds. A sample of 55 babies using the competitors brand revealed a mean increase of 8.8 pounds with a standard deviation of 2.9 pounds. At 95% level of confidence, can we conclude that the babies using the Gibbs product gained LESS weight?

10. What key words tell you that the Hypothesis is to be set up as a 1 Tail UPPER?

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#### Solution Preview

1. What is pooled variance and why is it important?

Assume we are conducting a hypothesis test to evaluate two sample means from independent samples, with the variance unknown, but we know it is the same for both populations. Then we use the pooled estimate of the variance given by [ (n1 - 1)s1^2 + (n2-1)s2^2 ] / (n1+n2-2)

2. Explain what interval data is and give an example:

Interval data is continuous data. When differences are interpretable, except wherever there is no "natural" zero. A good example is temperature in Fahrenheit degrees.
Ratios are insignificant for interval data. You cannot say, for example, that one day is double as hot as another day.
For Example , study of a seven thing graphic scale for three aspects of dyspnea (throat closing, chest tightness, and effort), children were asked to place these images on along a visual analog scale.

3. Write the formula for a problem that has 2 sample populations greater than 30 and the standard deviations are known and equal:
Solution:

The formula for a problem that has 2 sample populations greater than 30 and the standard deviations are known and equal is

4. Write the formula for pooled variance.

The formula for pooled variance is

5. Please analyze the following data:

A school district wants to determine if the girls are equal to boys in test scores. They took a random sample of 22 8th grade girls and 24 8th grade boys. The district looked at recent CAT scores and found the mean score for girls to be xbar1 = 25 and the mean score for boys to be xbar2 = 26. The standard deviation for the girls is s1 = 2.2 and for the boys the standard deviation is s2 = 3.4. Is the school district correct in assuming the girls are equal in performance to boys. They are 95% sure their assumption is correct.

Solution:

Given that
n1 = 228 xbar1 = 25 s1 = 2.2

n2 = 248 xbar2 = 26 s2 = 3.4.

Hypotheses:

Null Hypothesis:

That is, H0: The girls are equal in performance to boys

Alternative Hypothesis:

That is, H1: The girls are not equal in performance to boys

Level of Significance: α = 0.05

Test Statistic:
Since both the sample sizes n1 = 228 and n2 = 248 are greater than 30, we can use the large sample test Z-test to test the difference between the two means. The test statistic is given below:

The output is obtained using megastat

Hypothesis Test: Independent Groups (z-test)
Girls Boys
...

#### Solution Summary

The solution examines the importance of pooled variance. Interval data is explained and an example is provided.

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