# Recent questions in Confidence intervals

Confidence intervals

### Assume the sample is from a normally distributed population and construct the indicated confidence intervals for (a) the population variance $$\displaystyle\sigma^{{{2}}}$$ and (b) the population standard deviation $$\displaystyle\sigma$$. Interpret the results. The maximum wind speeds (in knots) of 13 randomly selected hurricanes that have hit the U.S. mainland are listed. Use a 95% level of confidence. $$\begin{matrix} 70 & 85 & 70 & 75 & 100 & 100 & 110 & 105 & 130 & 75 & 85 & 75 & 70 \end{matrix}$$

Confidence intervals

### Refer to the Business and Society (Mar. 2011) study on the sustainability behaviors of CPA corporations. Recall that the level of support for corporate sustainability (measured on a quantitative scale ranging from 0 to 160 points) was obtained for each in a sample of 992 senior managers at CPA firms. The accompanying MlNITAB printout gives 90% confidence intervals for both the variance and standard deviation of level of support for all senior managers at CPA firms. Statistics VariableNStDevVariance Support99226.9722 90% Confidence Intervals VariableMethodCI for StDevCI for Variance SupportChi-Square(25.9, 27.9)(672, 779) a. Locate the 90% confidence interval for σ2 on the printout. b. Use the sample variance on the printout to calculate the 90% confidence interval for σ2. Does your result agree with the interval shown on the printout? c. Locate the 90% confidence interval for a on the printout. d. Use the result, part a, to calculate the 90% confidence interval for a. Does your result agree with the interval shown on the printout? e. Give a practical interpretation of the 90% confidence interval for a. f. What assumption about the distribution of level of support is required for the inference, part e, to be valid? Is this assumption reasonably satisfied?

Confidence intervals

### Prove this version of the Bonferroni inequality: $$\displaystyle{P}{\left(\cap^{{{n}}}_{i=1}{A}_{{{i}}}\right)}\geq{1}-{\sum_{{{i}={1}}}^{{n}}}{P}{\left({{A}_{{{i}}}^{{{c}}}}\right.})$$(Use Venn diagrams if you wish.) In the context of simultaneous confidence intervals, what is Ai and what is $$\displaystyle{A}^{{{c}}}_{i}$$?

Confidence intervals

### In the article "On Judging the Significance of Differences by Examining the Overlap Between Confidence Intervals," by Schenker and Gentleman (American Statistician, Vol. 55, No. 3), the authors consider sample data in this statement: "Independent simple random samples, each of size 200, have been drawn, and 112 people in the first sample have the attribute, whereas 88 people in the second sample have the attribute." Based on the preceding results, what should you conclude about the equality of $$p_{1}$$ and $$p_{2}$$? Which of the three preceding methods is least effective in testing for the equality of $$p_{1}$$ and $$p_{2}$$?

Confidence intervals

### Assume the sample is from a normally distributed population and construct the indicated confidence intervals for (a) the population variance $$\sigma$$;2 and (b) the population standard deviation$$\sigma$$;. Interpret the results. The acceleration times (in seconds) from 0 to 60 miles per hour for 33 randomly selected sedans are listed. Use a 98% level of confidence. 6.5 5.0 5.2 3.3 6.6 6.3 5.1 5.3 5.4 9.5 7.5 4.5 5.8 8.6 6.9 8.1 6.0 6.7 7.9 8.8 7.1 7.9 7.2 18.4 9.1 6.8 12.5 4.2 7.1 9.9 9.5 2.8 4.9

Confidence intervals

### Consider two independent distributions that are mound-shaped. A random sample of size $$n_{1}=36$$ from the first distribution showed $$x_{1}=15$$, and a random sample of size $$n_{2}=40$$ from the second distribution showed $$x_{2}=14$$. Based on the confidence intervals you computed, can you be 95% confident that $$\mu_{1}$$ is larger than $$\mu_{2}$$? Explain.

Confidence intervals

### Find the levels of the confidence intervals that have the following values of $$\displaystyle{z}_{{\frac{\alpha}{{2}}}}$$: a. $$\displaystyle{z}_{{\frac{\alpha}{{2}}}}={1.96}$$ b. $$\displaystyle{z}_{{\frac{\alpha}{{2}}}}={2.17}$$ c. $$\displaystyle{z}_{{\frac{\alpha}{{2}}}}={1.28}$$ d. $$\displaystyle{z}_{{\frac{\alpha}{{2}}}}={3.28}$$

Confidence intervals

### Another type of confidence interval is called a one-sided confidence interval. A one-sided confidence interval provides either a lower confidence bound or an upper confidence bound for the parameter in question. You are asked to examine one-sided confidence intervals. Presuming that the assumptions for a one-mean z-interval are satisfied, we have the following formulas for (1−α)-level confidence bounds for a population mean $$\displaystyle\mu$$: Lower confidence bound: $$\bar{x}-z_{\alpha} \cdot \sigma/\sqrt{n}$$, Upper confidence bound: $$\bar{x}+z_{\alpha} \cdot \sigma/\sqrt{n}$$. Interpret the preceding formulas for lower and upper confidence bounds in words.

Confidence intervals

### Use the one-standard-deviation $$\displaystyle{x}^{{{2}}}$$-test and the one-standard-deviation $$\displaystyle{x}^{{{2}}}$$-interval procedure to conduct the required hypothesis test and obtain the specified confidence interval. $$s=2$$ and $$n=10$$ a. $$\displaystyle{H}_{{{0}}}:\sigma={4},{H}_{{{m}{a}{t}{h}{r}{m}{\left\lbrace{a}\right\rbrace}}}:\sigma{<}{4},\alpha={0.05}$$. b. $$90\%$$ confidence interval

Confidence intervals

### Suppose that you have obtained data by taking a random sample from a population and that you intend to find a confidence interval for the population mean, $$\mu$$. Which confidence level, 95% or 99%, will result in the confidence interval's giving a more precise estimate of $$\mu$$?

Confidence intervals

### Overproduction of uric acid in the body can be an indication of cell breakdown. This may be an advance indication of illness such as gout, leukemia, or lymphoma. Over a period of months, an adult male patient has taken eight blood tests for uric acid. The mean concentration was $$\bar{x}=5.35mg/dl$$. The distribution of uric acid in healthy adult males can be assumed to be normal with $$\sigma=1.85mg/dl\ \sigma=1.85mg/dl.$$ What conditions are necessary for your calculations?

Confidence intervals

### Let p be the population proportion for the situation. (a) Find point estimates of p and q, (b) construct 90% and 95% confidence intervals for p, and (c) interpret the results of part (b) and compare the widths of the confidence intervals. In a survey of 1035 U.S. adults, 745 say they want the U.S. to play a leading or major role in global affairs.

Confidence intervals

### Analyze the Transylvania effect data by calculating 95% Tukey confidence intervals for the pairwise differences among the admission rates for the three different phases of the moon. How do your conclusions agree with (or differ from) those already discussed? Let $$\displaystyle{Q}_{{{.05},{3},{22}}}={3.56}$$

Confidence intervals

### Use the information to construct 90% and 99% confidence intervals for the population mean. Interpret the results and compare the widths of the confidence intervals. A group of researchers estimates the mean length of time (in minutes) the average U.S. adult spends watching television using digital video recorders (DVRs) and other forms of time-shifted television each day. To do so, the group takes a random sample of 30 U.S. adults and obtains the times (in minutes) below. 29 12 23 24 33 24 28 31 18 27 27 32 17 13 17 12 21 32 26 16 28 28 21 24 29 13 20 13 21 27 From past studies, the research council assumes that σσ is 6.5 minutes.

Confidence intervals

### Use the one-standard-deviation $$\displaystyle{x}^{{{2}}}$$-test and the one-standard-deviation $$\displaystyle{x}^{{{2}}}$$-interval procedure to conduct the required hypothesis test and obtain the specified confidence interval. s=7 and n=26 a. $$\displaystyle{H}_{{{0}}}:\sigma={5},{H}_{{{m}{a}{t}{h}{r}{m}{\left\lbrace{a}\right\rbrace}}}:\sigma{>}{5},\alpha={0.01}$$. b. 98% confidence interval.

Confidence intervals

### Overproduction of uric acid in the body can be an indication of cell breakdown. This may be an advance indication of illness such as gout, leukemia, or lymphoma. Over a period of months, an adult male patient has taken eight blood tests for uric acid. The mean concentration was $$\bar{x}=5.35 \mathrm{mg} / \mathrm{dl.}$$ The distribution of uric acid in healthy adult males can be assumed to be normal, with $$\sigma=1.85 \mathrm{mg} / \mathrm{dl}$$ .Find a 95% confidence interval for the population mean concentration of uric acid in this patient's blood. What is the margin of error?

Confidence intervals

### Assume the sample is from a normally distributed population and construct the indicated confidence intervals for (a) the population variance $$\displaystyleσ^{{{2}}}$$ and (b) the population standard deviation σσ. Interpret the results. The diameters (in inches) of 18 randomly selected bolts produced by a machine are listed. Use a 95% level of confidence. 4.477 4.425 4.034 4.317 4.003 3.760 3.818 3.749 4.240 3.941 4.131 4.545 3.958 3.741 3.859 3.816 4.448 4.206

Confidence intervals

### Let p be the population proportion for the situation. (a) Find point estimates of p and q, (b) construct 90% and 95% confidence intervals for p, and (c) interpret the results of part (b) and compare the widths of the confidence intervals. In a survey of 2202 U.S. adults, 1167 think antibiotics are effective against viral infections.

Confidence intervals