The National Sleep Foundation surveyed representative samples of adults in six different countries to ask
questions about sleeping habits. Each person in a representative sample of 250 adults in each of these
countries was asked how much sleep they get on a typical work night. For the United States, the sample
mean was 391 minutes, and for Mexico the sample mean was 426 minutes.
The report also gave data for representative samples of 250 adults in Canada and 250 adults in England.
The sample mean amount of sleep on a work night was 423 minutes for the Canada sample and
409 minutes for the England sample. Suppose that the sample standard deviations were 38 minutes for the
Canada sample and 43 minutes for the England sample.
(a) Construct a 95% confidence interval estimate of the difference in the mean amount of sleep (in
minutes) on a work night for adults in Canada and adults in England. (Use Canada − England. Use
technology to calculate your critical value. Enter your answer using interval notation. Round your
numerical values to three decimal places.)
(b) Based on the confidence interval from part (a), would you conclude that there is evidence of a
difference in the mean amount of sleep on a work night for the two countries? Explain why or why
not
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To construct a confidence interval estimate of the difference in the mean amount of sleep for adults in Canada and adults in England, we can use the formula:
CI = (sample mean difference) ± (critical value) * (standard error)
(a) First, we need to calculate the standard error. The formula for standard error is:
standard error = sqrt[(standard deviation1^2 / sample size1) + (standard deviation2^2 / sample size2)]
For Canada:
standard deviation1 = 38 minutes
sample size1 = 250
For England:
standard deviation2 = 43 minutes
sample size2 = 250
Plugging in the values:
standard error = sqrt[(38^2 / 250) + (43^2 / 250)] = sqrt[1444/250] ≈ 9.548
Next, we need to calculate the critical value. Since the sample sizes are both 250, we can assume the sampling distribution of the mean difference is approximately normal. Looking up the critical value for a 95% confidence interval in a standard normal distribution table or using technology (such as a calculator or statistical software), we find the critical value to be approximately 1.96.
Now, we can calculate the confidence interval:
CI = (sample mean difference) ± (critical value) * (standard error)
sample mean difference = 423 - 409 = 14 minutes
CI = 14 ± 1.96 * 9.548 ≈ 14 ± 18.726 ≈ (-4.726, 32.726)
Therefore, the 95% confidence interval estimate for the difference in mean amount of sleep on a work night for adults in Canada and adults in England is approximately (-4.726, 32.726) minutes.
(b) Based on the confidence interval from part (a), we would conclude that there is evidence of a difference in the mean amount of sleep on a work night for the two countries. This is because the confidence interval does not include zero, which suggests that the mean amount of sleep for adults in Canada is likely to be different from the mean amount of sleep for adults in England.
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