Master Confidence Intervals: Examples, Equations, and Questions

A college lecturer randomly selected thirty-five students in his class to complete an assigned task andit is found that the 95% confidence interval for the true average time in completing the task is(17.278, 28.722) minutes. It is thought that this interval is too wide, and a confidence interval of totalwidth 5 minutes will be more desirable.(a) Using the same value of , find the new confidence interval in this case.(b) Find the confidence level for the interval in part (a).

A physician who has a group of thirty-eight female patients aged 18 to 24 on a special diet wishes to estimate the effect of the diet on total serum cholesterol. For this group, their average serum cholesterol is 188.4 (measured in mg/100mL). Because of a large-scale government study, the physician is willing to assume that the total serum cholesterol measurements are normally distributed with standard deviation of ${\displaystyle \sigma =40.7}$. Find a 95% confidence interval of the mean serum cholesterol of patients on the special diet. Does the diet seem to have any effect on their serum cholesterol, given that the national average for women aged 18 to 24 is 192.0?

A clarification on the level of confidence for a Chi-squared distribution
Suppose we have ${\displaystyle {\chi }_{1-\alpha ,29}^2=45.22}$, where the Chi-squared distribution has 29 degrees of freedom and ${\displaystyle (1-\alpha )\mathrm{\%}}$ level of confidence. How then do we compute ${\displaystyle 1-\alpha }$? The notation of ${\displaystyle {\chi }^2}$. I've found in books and online sources all come in the form of ${\displaystyle {\chi }_c^2}$, where c is the degrees of freedom.
I know this question seems trivial, but I'm still relatively new to Statistics.. Some clarification will be great!

A preliminary sample of 1200 engineers has been taken, out of which 260 are females. The sample is used to estimate the proportion of female engineers denoted by pF .
Estimate the 95% confidence interval for pF.

90% Confidence Interval for Difference Between Two Proportions
In random, independent samples of 225 adults and 250 teenagers who watched a certain television show, 99 adults and 125 teens indicated that they liked the show. Let ${\displaystyle p_1}$ be the proportion of all adults watching the show who liked it, and let ${\displaystyle p_2}$ be the proportion of all teens watching the show who liked it. Find a 90% confidence interval for ${\displaystyle p_1-p_2}$.
1. what is the lower limit of the 90% confidence interval
2. what is the upper limit of the 90% confidence interval

We wish to estimate what percent of adult residents in a certain county are parents. Out of 200 adult residents sampled, 176 had kids. Based on this, construct a 99% confidence interval for the proportion, p, of adult residents who are parents in this county.

Ten randomly selected people were asked how long they slept at night. The mean time was 7.1 hours, and the population variance was 0.6084 hour. Find the 99% confidence interval of the mean time. Assume the variable is normally distributed. In connection with the given data, explain the concept of the statement ‘the sample mean is significant or not significant at the given significance level’.

Here are summary statistic for randomly selected weights of newborn girls: ${\displaystyle n=211,\bar{x}=33.7hg,s=6.8hg.}$ Construct a confidence intercal estimate of the mean. Use a 90% confidence level. Are these results very different from the confidence interval ${\displaystyle 32.1hg<\mu <34.3hg}$ with only 16 sample values, ${\displaystyle \bar{x}=33.2hg}$, and ${\displaystyle s=2.6hg}$?
What is the confidence interval for the population mean ${\displaystyle \mu }$?
Are the results between the two confidence intervals very different?
a) Yes, because the confidence interval limits are not similar.
b) No, because each confidence interval contains the mean of the other confidence interval.
c) No, because the confidence interval limits are similar
d) Yes, because one confidence interval does not contain the mean of the other confidence interval.

Let us assume that 1000 different random samples was selected from the same population and for each of them a confidence interval of the population mean was constructed (using exactly the same method) with confidence level of 96%. On average, how many of those 1000 different confidence intervals would contain the true value of the population mean?
What would be the answer if confidence level was 99%?

In order to estimate the mean 30-year fixed mortgage rate for a home loan in the United States, a random sample of 25 recent loans is taken. The average calculated from this sample is 6.95%. It can be assumed that 30-year fixed mortgage rates are normally distributed with a standard deviation of 0.6%. Compute 90% and 99% confidence intervals for the population mean 30-year fixed mortgage rate.

Confidence intervals, again Several factors are involved in the creation of a confidence interval. Among them are the sample size, the level of confidence, and the margin of error. Which statements are true?
a) For a given sample size, reducing the margin of error will mean lower confidence.
b) For a certain confidence level, you can get a smaller margin of error by selecting a bigger sample.
c) For a fixed margin of error, smaller samples will mean lower confidence.
d) For a given confidence level, a sample 9 times as large will make a margin of error one third as big.

A recent survey of 7 social networking sites has a mean of 14.69 million visitors for a specifie month. The standard deviation was 4.4 millon. Find the 95% confidence interval of the true mean. Assume the variable is normally distributed. Round your answers to two decimal places.

The manufacturer of LED lights would like to include as part of their advertising the number of hours an LED is expected to work. A sample of 20 lights revealed the following number of hours worked.
$$\begin{array}{|cccccccccc|}\hline 2698& 2628& 2474& 2895& 2672& 2675& 2486& 2852& 2733& 2655\\ 2765& 2794& 2486& 2627& 2624& 2785& 2692& 2654& 2518& 2650\\ \hline\end{array}$$
Part 1: Draw normal probability plot for the above data and make comment.
Part 2: Assuming given data to be approximately normally distributed, Construct 90%, 95% and 99% confidence intervals for the mean number of hours that the LED lights of this brand has worked.
Part 3: Construct 90%, 95% and 99% confidence intervals for the standard deviation of the number of hours that the LED lights of this brand worked.

An article regarding interracial dating and marriage recently appeared in a newspaper. Of the 1713 randomly selected adults, 301 identified themselves as Latinos, 325 identified themselves as blacks, 255 identified themselves as Asians, and 778 identified themselves as whites. Among Asians, 79% would welcome a white person into their families, 71% would welcome a Latino, and 66% would welcome a black person.
If you are using a Student's t-distribution, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.)
Construct the 95% confidence intervals for the three Asian responses. (Round your answers to four decimal places.)
Construct the 95% confidence intervals for the three Asian responses. (Round your answers to four decimal places.)
a) Welcome a white peron.
b) welcome a black person.
c) welcome a latino

Determine (without solving the problem) an interval in which the solution of the given initial value problem is certain to exist. ${\displaystyle (t-3)y^{\prime }+\left(\ln t\right)y=2t,y\left(1\right)=2}$

In an experiment the sorption was measured for a variety of hazardous organic chloroalkane solvents. The sample measurements were; 1.18, 1.35, 0.57, 1.16, 1.12, 0.91, 0.83, 0.43 Assuming that measurements are normally distributed.
i) What is the mean of the sample for sorption of chloroalkane solvents.
ii) What is the standard error of the mean for the sample of organic chloroalkane solvents?
iii) What the margin of error of the 99% confidence intervals for the population mean of sorption of chloroalkane solvents.
iv) What is the lower limit for the 99% confidence intervals for the population mean of sorption of chloroalkane solvents.

A Bank wishes to determine the mean income of its credit card holders and more specifically wants to investigate whether there are any differences in the mean income between males and females. Confidence interval as a statistical inference method has been considered to respond this objective.
You are required to
1) Comment on the choice of confidence intervals as a statistical inference method for the above. Describe how you will proceed to determine the corresponding confidence intervals.
2) Explain how you would use confidence intervals as evidence suggesting that in average males and females income differ.
Use Numerical examples to illustrate as and when required.
(Word count around 250 words)

A random sample of 324 medical doctors showed that 164 had a solo practice.
a) Let p represent the proportion of all medical doctors who have a solo practice. Find a point estimate for p. (Use 3 decimal places.)
b) Find a 90% confidence interval for p. (Use 3 decimal places.)
lower limit
upper limit
c) Give a brief explanation of the meaning of the interval.
10% of the confidence intervals created using this method would include the true proportion of physicians with solo practices.
90% of all confidence intervals would include the true proportion of physicians with solo practices.
10% of all confidence intervals would include the true proportion of physicians with solo practices.
90% of the confidence intervals created using this method would include the true proportion of physicians with solo practices.

Which of the following statements is false?
1) We use one-sample procedures when our samples are equal in size but arent independent
2) Everything else being equal, a confidence interval based on 15 degrees of freedom will be narrower than one based on 10 degrees of freedom.
3) We can pool our estimates of the population variance ifwe want a narrower confidence interval for a given confidence level.
A. None of the above gives the correct response
B. 2 only
C. 1 only
D. 3 only
E. 1 and 2 only

Here are summary statistics fro randomly selected weights of newborn girs:
${\displaystyle n=240,\bar{x}=26.1hg,s=6.3hg}$
Construct a confidence interval estimate of the mean. Use a ${\displaystyle 99\mathrm{\%}}$ confidence level. Are these results very different from the confidence interval
${\displaystyle 25.1hg<\mu <28.1hg}$
with only 13 sample values,
${\displaystyle \bar{x}=26.6hg}$ and ${\displaystyle s=1.8hg?}$
What is the confidence interval for the population mean ${\displaystyle \mu ?}$
Are the results between the two confidence intervals very different?
a) No, because each confidence interval contains the mean of the other confidence interval.
b) No, because the confidence interval limits are similar.
c) Yes, because the confidence interval limits are not similar.
d) Yes, because one confidence interval does not contain the mean of the other confidence interval.