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7.4 SUMMARY WORKSHEET
How do we know when to use zα/2 or tα/2 ?
a) If you are working with a categorical variable (estimating a population proportion, p) always use zα/2.
b) If you are working with a quantitative variable (estimating a population mean, µ) and you DO know σ, use zα/2.
c) If you are working with a quantitative variable (estimating a population mean, µ) and you DO NOT know σ, use tα/2.
**Remember that the population distribution must be normal or n must be large for quantitative variables.**
Example 1: Assume we want to construct a confidence interval using the given confidence level. Do one of the following, as
appropriate: (i) Find the critical value zα/2, (ii) find the critical value tα/2, (iii) state that neither the normal nor the t distribution applies.
a) 95%; n = 34; σ is unknown; population appears to be normally distributed.
b) 90%; n = 72; σ is known; population appears to be normally distributed.
c) 98%; n = 22; σ is unknown; population appears to be very skewed.
d) 94%; n = 51; σ is unknown; population appears to be skewed.
e) 95%; n = 200; σ = 24.5; population appears to be skewed.
Example 2: A common claim is that garlic lowers cholesterol levels. In a test of the effectiveness of garlic, 46 subjects were treated
with doses of raw garlic, and their cholesterol levels were measured before and after the treatment. The changes in their levels of LDL
cholesterol (in mg/dL) have a mean of 3.2 and a standard deviation of 18.6.
a) What is the best point estimate of the population mean net change in LDL cholesterol after the garlic?
b) Construct a 95% confidence interval estimate of the mean net change in LDL cholesterol after the garlic treatment.
c) What does the confidence interval suggest about the effectiveness of garlic in reducing LDL cholesterol?
Name: ____________________________________
Example 3: In a study designed to test the effectiveness of Echinacea for treating upper respiratory tract infections in children, 401
children were treated with Echinacea and 301 other children were given a placebo. The numbers of days of peak severity of
symptoms for the Echinacea treatment group had a mean of 6.0 days and a standard deviation of 2.3 days. The numbers of days of
peak severity of symptoms for the placebo group had a mean of 6.1 days and a standard deviation of 2.4 days.
a) Construct the 95% confidence interval for the mean number of days of peak severity of symptoms for those who receive Echinacea
treatment.
b) Construct the 95% confidence interval for the mean number of days of peak severity of symptoms for those who are given a
placebo.
Example 4: In a study designed to test the effectiveness of magnets for treating back pain, 20 patients were given a treatment with
magnets and also a sham treatment without magnets. Pain was measured using a standard Visual Analog Scale (VAS). After given
the magnet treatments, the 20 patients had VAS scores with a mean of 5.0and a standard deviation of 2.4. After being given the sham
treatments, the 20 patients had VAS scores with a mean of 4.7 and a standard deviation of 2.9.
a) Construct the 95% confidence interval estimate of the mean VAS score for patients given the magnet treatment.
b) Construct the 95% confidence interval estimate of the mean VAS score for patients given the sham treatment.
Example 5: A simple random sample of 40 statistics students at OHS participated in an experiment to test their ability to determine
when 1 min has passed. The students yielded a sample mean of 58.3 sec. Assume that the perception time of all statistics students has
a standard deviation of 9.5 sec.
a) Construct a 95% confidence interval estimate of the mean time.
b) Construct a 99% confidence interval estimate of the mean time.
c) Which of the preceding confidence intervals is wider? Why?