Master Confidence Intervals: Examples, Equations, and Questions

A clinical trial was conducted to test the effectiveness of a drug for treating insomnia in older subjects. Before​ treatment, 13 subjects had a mean wake time of 101.0 min. After​ treatment, the 13 subjects had a mean wake time of 94.6 min and a standard deviation of 24.9 min. Assume that the 13 sample values appear to be from a normally distributed population and construct a ${\displaystyle 95\mathrm{\%}}$ confidence interval estimate of the mean wake time for a population with drug treatments. What does the result suggest about the mean wake time of 101.0 min before the​ treatment? Does the drug appear to be​ effective?
Construct the ${\displaystyle 95\mathrm{\%}}$ confidence interval estimate of the mean wake time for a population with the treatment.
$min<\mu <min$ ​(Round to one decimal place as​ needed.)
What does the result suggest about the mean wake time of 101.0 min before the​ treatment? Does the drug appear to be​ effective?
The confidence interval ▼ does not include| includes the mean wake time of 101.0 min before the​ treatment, so the means before and after the treatment ▼ could be the same |are different. This result suggests that the drug treatment ▼ does not have | has a significant effect.

The scores of a certain population on the Wechsler Intelligence Scale for children (WISC) are thought to be normally distributed with a mean and standard deviation of 10. A simple random sample of twenty -five children from this population is taken and each is given the WISC. The mean of the twenty-five score is ${\displaystyle \bar{x}=104.32}$. Based on these data a ${\displaystyle 95\mathrm{\%}}$ confidence interval for ${\displaystyle \mu }$ is computed.
The ${\displaystyle 95\mathrm{\%}}$ confidence interval for ${\displaystyle \mu }$ is?

Customers randomly selected at a grocery store included 172 women and 45 men. 74 of the women and 12 of the men used coupons. Find the ${\displaystyle 95\mathrm{\%}}$ confidence interval.

Unoccupied seats on flights cause airlines to lose revenue. Suppose a large airline wants to estimate its average number of unoccupied seats per flight over the past year. To accomplish this, the records of 200 flights are randomly selected and the number of unoccupied seats is noted for each of the sampled flights. The sample mean is 11.6 seats and the sample standard deviation is 4.3 seats. ${\displaystyle s_x=?,n=?,n-1=?}$. Construct a ${\displaystyle 92\mathrm{\%}}$ confidence interval for the population average number of unoccupied seats per flight. (State the confidence interval. (Round your answers to two decimal places.)

Your client from question 4 decides they want a ${\displaystyle 90\mathrm{\%}}$ confidence interval instead of a ${\displaystyle 95\mathrm{\%}}$ one. To remind you - you sampled 36 students to find out how many hours they spent online last week. Your sample mean is 46. The population standard deviation, ${\displaystyle \sigma }$, is 7.4. What is the ${\displaystyle 90\mathrm{\%}}$ confidence interval for this sample mean? Keep at least three decimal places on any intermediate calculations, then give your answer rounded to 1 decimal place, with the lower end first then upper.
(_______ , ___________ )

Assume that the true value, that is the population mean, the average height of french women in france is 167.5 cm. Students in a class of 160 students each took a random sample of french women of size 100, measured the height of the women and calculated a 95% confidence interval (confidence interval) for the average height.How many students can be expected to calculate a confidence interval that does not include the number 167.5?
15 8 5 20 1

A clinical trial was conducted to test the effectiveness of a drug for treating insomnia in older subjects. Before treatment, 24 subjects had a mean wake time of 104.0 min. After treatment, the 24 subjects had a mean wake time of 94.5 min and a standard deviation of 23.2 min. Assume that the 24 sample values appear to be from a normally distributed population and construct a ${\displaystyle 99\mathrm{\%}}$ confidence interval estimate of the mean wake time for a population with drug treatments. What does the result suggest about the mean wake time of 104.0 min before the treatment? Does the drug appear to be effective?
Construct the ${\displaystyle 99\mathrm{\%}}$ confidence interval estimate of the mean wake time for a population with the treatment.
${\displaystyle ?min<\mu <?min}$

a. Locate the critical points of ƒ.
b. Use the First Derivative Test to locate the local maximum and minimum values.
c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist).
${\displaystyle f\left(x\right)=\frac{x^2}{x^2-1}}$ on ${\displaystyle [-4,4]}$

Explain what is the test statistic used for constructing confidence interval on a proportion.

Based on a simple random sample of 1300 college students, it is found that 299 students own a car. We wish to construct a ${\displaystyle 90\mathrm{\%}}$ confidence interval to estimate the proportions ? of all college students who own a car.
A) Read carefully the text and provide each of the following:
The sample size ${\displaystyle ?=}$
from the sample, the number of college students who own a car is ${\displaystyle ?=}$
the confidence level is ${\displaystyle CL=}$ ${\displaystyle \mathrm{\%}}$.
B) Find the sample proportion ${\displaystyle \widehat{?}=}$
and ${\displaystyle \widehat{?}=1-\widehat{?}=}$

We walk from Central Park to Empire State Building with a constant speed. Our speed X is a random variable uniformly distributed between 3 and 4.5 km/h. The distance is 3 km. Calculate the probability distribution of the total walking time T:
a. Calculate the CDF of T from the distribution of X.
b. Calculate the PDF of T from irs CDF.
Don't forget to specify the intervals for these functions explicitly.

Express the confidence interval ${\displaystyle (0.412,0.789)}$ in the form of ${\displaystyle \hat{p}\pm E}$.

From a simple random sample of 14 shoppers surveyed, the sample mean of money spent for holiday gifts was 145 dollars with a sample standard deviation of 11 dollars. We wish to construct a ${\displaystyle 99\mathrm{\%}}$ confidence interval for the population mean of money spent for holiday gifts.
Read carefully the text and provide each of the following:
The sample size ${\displaystyle ?=}$,
the sample mean ${\displaystyle ?\equiv }$,
the sample standard deviation ${\displaystyle ?=}$, NKS the confidence level ${\displaystyle CL=\mathrm{\%}}$.

Find the value:
${\displaystyle t_{\frac{a}{2}}}$
${\displaystyle n=23}$ for the ${\displaystyle 95\mathrm{\%}}$ confidence interval for the mean.

find the value:
${\displaystyle t_{\frac{a}{2}}}$
${\displaystyle n=10}$ for the ${\displaystyle 90\mathrm{\%}}$ confidence interval for the mean.

Find ${\displaystyle t_{\frac{\alpha }{2}},n=15}$ for the ${\displaystyle 98\mathrm{\%}}$ confidence interval for the mean.

Determine the sample size taken from a normal distribution N(M, 25) to get length of interval of 4 using ${\displaystyle 90\mathrm{\%}}$ confidence level for the mean.

A newsgroup is interested in constructing a 90% confidence interval for the proportion of all Americans who are in favor of a new Green initiative. Of the 536 randomly selected Americans surveyed, 441 were in favor of the initiative. Round answers to 4 decimal places where possible. a. With 90% confidence the proportion of all Americans who favor the new Green initiative is between and
b. If many groups of 536 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.

A salesperson goes door-to-door in a residential area to demonstrate the use of a new household appliance to potential customers. At the end of a demonstration,the probability that the potential customer would place an order for the product is aconstant 0.2107. To perform satisfactorily on the job, the salesperson needs at leastfour orders. Assume that each demonstration is a Bernoulli trial.
a. If the salesperson wants to be at least ${\displaystyle 90\mathrm{\%}}$ confident of getting at least 4 orders, at least how many demonstrations should she make?
b. The salesperson has time to make only 22 demonstrations, and she still wants to be at least ${\displaystyle 90\mathrm{\%}}$ confident of getting at least 4 orders. She intends to gain this confidence by improving the quality of her demonstration and thereby improving the chances of getting an order at the end of a demonstration. At least to what value should this probability be increased in order to gain the desired confidence? Your answer should be accurate to four decimal places.

The number of points TEDU basketball team scores against Stanford basketball team is normally distributed. Suppose that over the last 10 games between the two teams TEDU scored the points given below:
a) Construct a ${\displaystyle 95\mathrm{\%}}$ two-sided confidence interval for the mean of the number of points scored by TEDU.
b) The coach of the TEDU team finds out that the variance of the number of points scored is 25. Given this new information, construct a ${\displaystyle 95\mathrm{\%}}$ two-sided confidence interval for the mean of the number of points scored by TEDU.
c) Compare the confidence intervals found in part (a) and (b). Explain the reason behind the difference.