problem set 5

1. Newborn weight. A study takes an SRS from a population of full-term infants. The standard deviation of birth weights in this population is 2 pounds. Calculate 95% confidence intervals for μ for samples in which:

a)      n = 81 and = 6.1 pounds

b)      n = 36 and = 7.0 pounds

c)       n = 9 and = 5.8 pounds

1. SIDS. A sample of 49 sudden infant death syndrome (SIDS) cases had a mean birth weight of 2998 g. Based on other births in the county, we will assume σ = 800 g. Calculate the 95% confidence interval for the mean birth weight of SIDS cases in the county. Interpret your results.

1. Hemoglobin. Hemoglobin levels in 11-year-old boys vary according to a Normal distribution with σ = 1.2 g/dL. (a) How large a sample is needed to estimate mean μ with 95% confidence so the margin of error is no greater than 0.5 g/dL? (b) How large a sample is needed to estimate μ with margin of error 0.5 g/dL with 99% confidence?

1. P-value and confidence interval. A two-sided test of H0: μ = 0 yields a P-value of 0.03. Will the 95% confidence interval for μ include 0 in its midst? Will the 99% confidence interval for μ include 0? Explain your reasoning in each instance.

1. Critical values for a t-statistic. The term critical value is often used to refer to the value of a test statistic that determines statistical significance at some fixed α level for a test. For example, ±1.96 are the critical values for a two-tailed z-test at α = 0.05.

a)      In performing a t-test based on 21 observations, what are the critical values for a one-tailed test when α = 0.05? That is, what values of the tstat will give a one-sided P-value that is less than or equal to 0.05? What are the critical values for a two-tailed test at α = 0.05?

1. Menstrual cycle length. Menstrual cycle lengths (days) in an SRS of nine women are as follows: {31, 28, 26, 24, 29, 33, 25, 26, 28}. Use this data to test whether mean menstrual cycle length differs significantly from a lunar month. (A lunar month is 29.5 days.) Assume that population values vary according to a Normal distribution. Use a two-sided alternative. Show all hypothesis-testing steps.

1. Menstrual cycle length. Exercise 6 calculated the mean length of menstrual cycles in an SRS of n = 9 women. The data revealed days with standard deviation s = 2.906 days.

a)      Calculate a 95% confidence interval for the mean menstrual cycle length.

b)      Based on the confidence interval you just calculated, is the mean menstrual cycle length significantly different from 28.5 days at α = 0.05 (two sided)? Is it significantly different from μ = 30 days at the same α-level? Explain your reasoning. (Section 10.4 in your text considered the relationship between confidence intervals and significance tests. The same rules apply here.)

1. Water fluoridation. A study looked at the number of cavity-free children per 100 in 16 North American cities BEFORE and AFTER public water fluoridation projects. The table below lists the data. You will need to manually type the data into StatCrunch to use that tool to calculate the requested information.

a)      Calculate delta values for each city. Then construct a stemplot of these differences. Interpret your plot.

b)      What percentage of cities showed an improvement in their cavity-free rate?

c)       Estimate the mean change with 95% confidence.