Master Confidence Intervals: Examples, Equations, and Questions

You are given the sample mean and the population standard deviation. Use this information to construct the​ 90% and​ 95% confidence intervals for the population mean. Interpret the results and compare the widths of the confidence intervals. From a random sample of 58 ​dates, the mean record high daily temperature in a certain city has a mean of ${\displaystyle {83.43}^{\circ }F}$. Assume the population standard deviation is ${\displaystyle {14.02}^{\circ }F}$. ${\displaystyle 90\mathrm{\%}=}$ ${\displaystyle 95\mathrm{\%}=}$ Which interval is wider?

You randomly select and weigh 27 samples of an allergy medication. The sample standard deviation is 1.22 milligrams. Assuming the weights are normally distributed, construct $90\mathrm{\%}$ confidence intervals for the population variance and standard deviation. Please interpret your result.

Use the Composite Trapezoidal, Simpson's and Midpoint Rules to approximate the integral
${\int }_1^{2}x\ln (x)dx,n=4$
(For the Midpoint Rule use $n+2$ sub intervals.)

A machine produces parts with lengths that are normally distributed with ${\displaystyle \sigma =0.66}$. A sample of 20 parts has a mean length of 75.98.
a)
Give a point estimate for ${\displaystyle \mu }$. (Give your answer correct to two decimal places.)
b)
Find the $95\mathrm{\%}$ confidence maximum error of estimate for ${\displaystyle \mu }$. (Give your answer correct to two decimal places.)
c)
Find the $95\mathrm{\%}$ confidence interval for ${\displaystyle \mu }$. (Give your answer correct to two decimal places.)
Lower limit ${\displaystyle =?}$
Upper limit ${\displaystyle =?}$

In replicate analyses, the carbohydrate content of a glycoprotein ( a protein with sugars attached to it) is found to be 12.6, 11.9, 13.0, 12.7, and 12.5 g of carbohydrate per 100 g of protein. Find the 90% confidence intervals for the carbohydrate content.

Solve the equation. Answer the following questions:
(a) Over what x - intervals is f increasing//decreasing?
(b) Find all critical points of f and test each for local maximum and local minimum.
(c) Over what x - intervals is f concave up//down?
$f(x)=4+8x^3-x^4$

A random sample of size ${\displaystyle n=25}$ from a normal distribution with mean ${\displaystyle \mu }$ and variance 36 has sample mean ${\displaystyle \bar{X}=16.3}$
a)
Calculate confidence intervals for ${\displaystyle \mu }$ at three levels of confidence: $80$. How to the widths of the confidence intervals change?
b) How would the CI width change if n is increased to 100?

A random sample of 5805 physicians in Colorado showed that 3332 provided at least some charity care (i.e., treated poor people at no cost).
a)
Let p represent the proportion of all Colorado physicians who provide some charity care. Find a point estimate for p. (Round your answer to four decimal places.)
b)
Find a $99\mathrm{\%}$ confidence interval for p. (Round your answer to three decimal places.)
lower limit ${\displaystyle =?}$
upper limit ${\displaystyle =?}$
c)
Is the normal approximation to the binomial justified in this problem? Explain

Find vert and horz asymptotes, and points of inflection for
$f(x)=\ln (x)/x$
The graph the function

In a survey of 2566 adults in a recent​ year, 1420 say they have made a New​ Years

A newsgroup is interested in constructing a $90\mathrm{\%}$ confidence interval for the proportion of all Americans who are in favor of a new Green initiative. Of the 555 randomly selected Americans surveyed, 437 were in favor of the initiative. Round answers to 4 decimal places where possible.
a)
With $90\mathrm{\%}$ confidence the proportion of all Americans who favor the new Green initiative is between ____ and ____.
b)
If many groups of 555 randomly selected Americans were surveyed, then a different confidence interval would be produced from each group.
About ____ percent of these confidence intervals will contain the true population proportion of Americans who favor the Green initiative and about ____ percent will not contain the true population proportion.

Consider the following.
$f(x)=49-x^2$
from $x=1tox=7,4$ subintervals
(a) Approximate the area under the curve over the specified interval by using the indicated number of subintervals (or rectangles) and evaluating the function at the right-hand endpoints of the subintervals.
(b) Approximate the area under the curve by evaluating the function at the left-hand endpoints of the subintervals.

Do teachers find their work rewarding and satisfying? The article presents the results of a survey of 399 primary school teachers and 264 senior teachers. Of the elementary school teachers, 223 said they were very satisfied with their jobs, whereas 127 of the high school teachers were very satisfied with their work. Estimate the difference between the proportion of all elementary school teachers who are satisfied and all high school teachers who are satisfied by calculating a ${\displaystyle 95\mathrm{\%}CI.(\text{Use}{P}_{\text{elementary}}-P_{\text{high}}}$ school. Round your answers to four decimal places.)

(a) The companys

Consider a particle moving along the x-axis where x(t) is the position of the particle at time t, x'(t) is its velocity, and x"(t) is its acceleration.
$x(t)=t^3-6t^2+9t-9,$
$0\le t\le 10$
a) Find the velocity and acceleration of the particle?
$x^{\prime }(t)=?$
$x"(t)=?$
b) Find the open t-intervals on which the particle is moving to the right?
c) Find the velocity of the particle when the acceleration is 0?

A researcher is interested in finding a $90\mathrm{\%}$ confidence interval for the mean number minutes students are concentrating on their professor during a one hour statistics lecture. The study included 117 students who averaged 40.9 minutes concentrating on their professor during the hour lecture. The standard deviation was 11.8 minutes. Round answers to 3 decimal places where possible.
a.
To compute the confidence interval use a ? distribution.
b.
With $90\mathrm{\%}$ confidence the population mean minutes of concentration is between ____ and ____ minutes.
c.
If many groups of 117 randomly selected students are studied, then a different confidence interval would be produced from each group. About ____ percent of these confidence intervals will contain the true population mean minutes of concentration and about ____ percent will not contain the true population mean number of minutes of concentration.

Find the future value of an annuity of $ 1300 paid at the end of each year for 10 years, if interest is earned at a rate of $7\mathrm{\%},$ compounded annually. (Round your answer to the nearest cent.)

Construct 98% confidence intervals for the a) population variance and the b)population standard deviation ${\displaystyle \sigma }$ The number of hours of reserve capacity of 10 randomly selected automotive batteries. 1.71, 1.87, 1.58, 1.61, 1.78, 1.98, 1.36, 1.58, 1.47, 2.05, Assume the sample is taken from a normally distributed population.

Solve and answer this question Are the results between the two confidence intervals very different
$27.2hg<\mu <29.4hg$
What is the confidence interval for the population mean $\mu ?$

Here are summary stastistics for randomly selected weights of newborn girls: ${\displaystyle n=224,\bar{x}=28.3\text{hg},s=7.1\text{hg}}$. Construct a confidence interval estimate of mean. Use a 98% confidence level. Are these results very different from the confidence interval ${\displaystyle 26.5\text{hg}<\mu <30.7\text{hg}}$ with only 14 sample values, ${\displaystyle \bar{x}=28.6}$ hg, and ${\displaystyle s=2.9}$ hg? What is the confidence interval for the population mean ${\displaystyle \mu }$?

$27,2hg<\mu <29,4hg?$ (Round to one decimal place as needed).

Are the results between the two confidence intervals very different?