Master Confidence Intervals: Examples, Equations, and Questions

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 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

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 σ. Interpret the results. The reserve capacities (in hours) of 18 randomly selected automotive batteries have a sample standard deviation of 0.25 hour. Use an 80% level of confidence.

For each of the following confidence intervals for the difference between two means,
$({\mu }_1-{\mu }_2)$
which mean is significantly larger?
a.(-10.5)

b.(-10,-5)

c.(5,10)

The file sapphire lists observed values of Young’s modulus (g) measured at various temperatures (T ) for sapphire rods (Ku 1969). Fit a linear relationship ${\displaystyle g={\beta }_0+{\beta }_{1}t}$, and form confidence intervals for the coefficients. Examine the residuals.

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. $s=6$ and $n=26$
a. ${\displaystyle H_0:\sigma =5,H_{\mathrm{a}}:\sigma >5,\alpha =0.01}$.
b. 98% confidence interval.

Is there a relationship between confidence intervals and two-tailed hypothesis tests? Let e be the level of confidence used to construct a confidence interval from sample data. Let $\alpha$ be the level of significance for a two-tailed hypothesis test. The following statement applies to hypothesis tests of the mean. (A corresponding relationship between confidence intervals and two-tailed hypothesis tests also is valid for other parameters such as p, $m{u}_1-{\mu }_2$, or $p_1-p_2$, which we will study in Section 9.3, 10.2, and 10.3.) Whenever the value of k given in the null hypothesis falls outside the ${\displaystyle c=1-\alpha }$ confidence interval for the parameter, we reject $H_0$. For example, consider a two-tailed hypothesis test with $\alpha =0.01\text{and}{H}_0:\mu =20H_1:\mu 20$ sample mean ${\displaystyle x¯=22}$ from a population with standard deviation ${\displaystyle \sigma =4}$.
(a) What is the value of c${\displaystyle c=1-\alpha }$;. Using the methods, construct a ${\displaystyle c=1-\alpha }$; confidence interval for μ from the sample data. What is the value of $\mu$; given in the null hypothesis (i.e., what is k)? Is this value in the confidence interval? Do we reject or fail to reject $H_0$ based on this information?
(b) using methods, find the P-value for the hypothesis test. Do we reject or fail to reject $H_0$? Compare your result to that of part (a).

People with high blood pressure suffer from hypertension. A study of the lipid profiles of hypertensive patients was carried out and the results published in Biology and Medicine (Vol. 2, 2010). Data on fasting blood sugar (milligrams/deciliter) and magnesium (milligrams/deciliter) in blood specimens collected from 50 patients diagnosed with hypertension were collected. The accompanying MlNlTAB printout gives 90% confidence intervals for the mean fasting blood sugar (FBS) and mean magnesium level (MAG). a. Locate and interpret the 90% confidence interval for mean fasting blood sugar on the printout. b. Locate and interpret the 90% confidence interval for mean magnesium level on the printout. c. If the confidence level is increased to 95%, what will happen to the width of the intervals? d. If the sample of hypertensive patients is increased from 50 to 100, what will likely happen to the width of the intervals?$\begin{array}{cccccc}\text{Variable}& \text{N}& \text{Mean}& \text{StDev}& \text{SE Mean}& \text{90\% CI}\\ \text{FBS}& \text{50}& \text{101.62}& \text{34.13}& \text{4.83}& \text{(93.53, 109.71)}\\ \text{MAG}& \text{50}& \text{1.94100}& \text{0.05607}& \text{0.00793}& \text{(1.92771, 1.95429)}\end{array}$

Construct the indicated confidence intervals for (a) the population variance 
${\displaystyle {\sigma }^2}$ 
and (b) the population standard deviation 
${\displaystyle \sigma }$ 
Assume that the population in the sample has a normal distribution.
${\displaystyle c=0.95,s^2=11.56,n=30}$

You are given the sample mean and the population standard deviation. Create 90% and 95% confidence intervals for the population mean using this information. Interpret the results and compare the widths of the confidence intervals. From a random sample of 36 business days from February 24, 2016, through February 24, 2017, the mean closing price of Apple stock was $116.16. Assume the standard deviation for the population is $10.27.