# Question

Consider the following questions about Confidence Intervals (CI).

A researcher tests Emotional Intelligence (EI) for a random sample of children selected from a population of all students who are enrolled in a school for gifted children. The researcher wants to estimate the mean EI for the entire school.

Let’s suppose that a researcher wants to set up a 95% CI for IQ scores using the following information. The population standard deviation σ for EI is not known.

The sample mean, M, is M = 130.

The sample standard deviation, s = 15.

The sample size, N, is N = 120.

The df = N – 1 = 119

Because the df is 119, the “tcritical” values that are used to indicate the “distance from the mean” that corresponds to the middle 95% of the sampling distribution are values of -1.96 (for the lower limit) and +1.96 (for the upper limit of the CI). See the table of critical t values in Appendix B. Use the df in the table that is closest to the df in the problem (in this case, df = 100 in the table is the closest to df = 119 in the problem).

The standard error of the mean, SEM, is computed by taking s divided by the square root of N.

To set up the 95% CI, the researcher uses these equations:

Lower limit = M – tcritical * SEM

Upper limit = M + tcritical * SEM

For the values given above, the limits of the 95% CI are:

Lower limit = 130 – 1.96* 1.37 = 127.31

Upper limit = 130 + 1.96* 1.37 = 132.69

Now let’s “experiment” to see how changing some of the values involved in computing the CI influences the width of the CI

Recalculate the Confidence Interval above to see how the lower and upper limits (and the width of the CI) change, as you vary the N in the sample (and leave all the other values the same).

a. What are the upper and lower limits of the CI and the width of the 95% CI if all of the other values remain the same (M = 130, s = 15), but you change the value of N to 16

b. What are the upper and lower limits of the CI and the width of the 95% CI if all of the values remain the same, but you change the value of N to 25?

c. What are the upper and lower limits of the CI and the width of the 95% CI if all of the other values remain the same (M = 130, s = 15), but you change the value of N to 49?

d. Based on the numbers you reported for Ns of 16, 25, and 49: How does the width of the CI change as the N (number of cases in the sample) increases?

e. What are the upper and lower limits, and the width of this CI, if you change the “Confidence” level to 80% (and continue to use M = 130, s= 15 and N = 49)

f. What are the upper and lower limits, and the width of the CI, if you change the confidence level to 99%? (Continue to use M = 130, s = 15 and N = 49)

g. How does increasing the level of “Confidence” from 80% to 99% affect the width of the CI?

A researcher tests Emotional Intelligence (EI) for a random sample of children selected from a population of all students who are enrolled in a school for gifted children. The researcher wants to estimate the mean EI for the entire school.

Let’s suppose that a researcher wants to set up a 95% CI for IQ scores using the following information. The population standard deviation σ for EI is not known.

The sample mean, M, is M = 130.

The sample standard deviation, s = 15.

The sample size, N, is N = 120.

The df = N – 1 = 119

Because the df is 119, the “tcritical” values that are used to indicate the “distance from the mean” that corresponds to the middle 95% of the sampling distribution are values of -1.96 (for the lower limit) and +1.96 (for the upper limit of the CI). See the table of critical t values in Appendix B. Use the df in the table that is closest to the df in the problem (in this case, df = 100 in the table is the closest to df = 119 in the problem).

The standard error of the mean, SEM, is computed by taking s divided by the square root of N.

To set up the 95% CI, the researcher uses these equations:

Lower limit = M – tcritical * SEM

Upper limit = M + tcritical * SEM

For the values given above, the limits of the 95% CI are:

Lower limit = 130 – 1.96* 1.37 = 127.31

Upper limit = 130 + 1.96* 1.37 = 132.69

Now let’s “experiment” to see how changing some of the values involved in computing the CI influences the width of the CI

Recalculate the Confidence Interval above to see how the lower and upper limits (and the width of the CI) change, as you vary the N in the sample (and leave all the other values the same).

a. What are the upper and lower limits of the CI and the width of the 95% CI if all of the other values remain the same (M = 130, s = 15), but you change the value of N to 16

b. What are the upper and lower limits of the CI and the width of the 95% CI if all of the values remain the same, but you change the value of N to 25?

c. What are the upper and lower limits of the CI and the width of the 95% CI if all of the other values remain the same (M = 130, s = 15), but you change the value of N to 49?

d. Based on the numbers you reported for Ns of 16, 25, and 49: How does the width of the CI change as the N (number of cases in the sample) increases?

e. What are the upper and lower limits, and the width of this CI, if you change the “Confidence” level to 80% (and continue to use M = 130, s= 15 and N = 49)

f. What are the upper and lower limits, and the width of the CI, if you change the confidence level to 99%? (Continue to use M = 130, s = 15 and N = 49)

g. How does increasing the level of “Confidence” from 80% to 99% affect the width of the CI?

## Answer to relevant Questions

Under what circumstances will the value of SS equal 0? Can SS ever be negative? Why do we divide by (N -1) rather than N when we compute a variance from an SS term? What is a sampling distribution? What do we know about the shape and characteristics of the sampling distribution for M, the sample mean? What recommendations did the APA task force make about reporting statistical results (are significance tests alone sufficient?) In your own words (not a direct quote from the chapter): what does it mean to say “p < .05”? What do you need to look for in bivariate screening (for each combination of categorical and quantitative variables)?Post your question

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