Master Confidence Intervals: Examples, Equations, and Questions

The monthly incomes for 12 randomly selected​ people, each with a​ bachelor's degree in​ economics, are shown on the right. Complete parts​ (a) through​ (c) below.
Assume the population is normally distributed.
a) Find the sample mean.
b) Find the sample standard deviation
c) Construct a ${\displaystyle 95\mathrm{\%}}$ confidence interval for the population mean ${\displaystyle \mu }$. A ${\displaystyle 95\mathrm{\%}}$ confidence interval for the population mean is
$\begin{array}{|ccc|}\hline 4450.42& 4596.96& 4366.46\\ 4455.62& 4151.52& 3727.77\\ 4283.26& 4527.94& 4407.68\\ 3946.49& 4023.61& 4221.73\\ \hline\end{array}$

Two different analytical tests can be used to determine the impurity level in steel alloys. Nine specimens are tested using both procedures, and the results are shown in the following tabulation.
$\begin{array}{|ccc|}\hline Specimen& Test-1& Test-2\\ 1& 1.2& 1.4\\ 2& 1.6& 1.7\\ 3& 1.5& 1.5\\ 4& 1.4& 1.3\\ 5& 1.8& 2.0\\ 6& 1.8& 2.1\\ 7& 1.4& 1.7\\ 8& 1.5& 1.6\\ 9& 1.4& 1.5\\ \hline\end{array}$
Find the ${\displaystyle 95\mathrm{\%}}$ confidence interval for the mean difference of the tests, and explain it.

Express the confidence interval $57.7\mathrm{\%}\pm 4.9\mathrm{\%}$ in interval form.
Express the answer in decimal format (donot eter as percents).

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. $\begin{array}{ccccccccccccc}70& 85& 70& 75& 100& 100& 110& 105& 130& 75& 85& 75& 70\end{array}$

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

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

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 $p_2$?

Take into account two mound-shaped independent distributions. 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 calculated confidence intervals, can you be 95% confident that ${\mu }_1$ is larger than ${\mu }_2$? Explain.

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. ${\displaystyle z_{\frac{\alpha }{2}}=3.28}$

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: $\overline{x}-z_{\alpha }\cdot \sigma /\sqrt{n}$, Upper confidence bound: $\overline{x}+z_{\alpha }\cdot \sigma /\sqrt{n}$. Interpret the preceding formulas for lower and upper confidence bounds in words.

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. ${\displaystyle H_0:\sigma =4,H_{\left\{a\right\}}:\sigma <4,\alpha =0.05}$.
b. $90\mathrm{\%}$ confidence interval

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 $\overline{x}=5.35mg/dl$. 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?

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 $\mu$?

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 ${\displaystyle Q_{.05,3,22}=3.56}$

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. Researchers have calculated the average daily time (in minutes) that Americans spend watching television using time-shifting devices like digital video recorders (DVRs) and other devices. 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.

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 $\overline{x}=5.35\mathrm{mg}/\mathrm{dl}.$ 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?