Master Confidence Intervals: Examples, Equations, and Questions

Express the confidence intervel (0.501, 0.609) in the from of ${\displaystyle p+E}$

A simple random sample of 60 items resulted in a sample mean of 80. The population standard deviation is ${\displaystyle \sigma =15}$
a) Compute the ${\displaystyle 95\mathrm{\%}}$ confidence interval for the population mean. Round your answers to one decimal place.
b) Assume that the same sample mean was obtained from a sample of 120 items. Provide a ${\displaystyle 95\mathrm{\%}}$ confidence interval for the population mean. Round your answers to two decimal places.
c) What is the effect of a larger sample size on the interval estimate?
Larger sample provides a-Select your answer-largersmallerItem 5 margin of error.

Find the critical value z a/2 that corresponds to a ${\displaystyle 93\mathrm{\%}}$ confidence level

If ${\displaystyle n=600}$ and ${\displaystyle \hat{p}=0.95}$, construct a ${\displaystyle 95\mathrm{\%}}$ confidence interval.
Give your answer to three decimals.
$<p<$

Survey A asked 1000 people how they liked the new movie Avengers: Endgame and ${\displaystyle 88\mathrm{\%}}$ said they did enjoy it.
Survey B also concluded that ${\displaystyle 82\mathrm{\%}}$ of people liked the move but they asked 1600 total moviegoers.
Which of the following is true about this comparison?
1. The margin of error is the same for both.
2. The confidence interval is smaller for Survey B.
3. Increasing the number of people asked does not change the ${\displaystyle 95\mathrm{\%}}$ confidence interval.
4. Survey A is more accurate since the percentage is higher.
5. Survey A has more approvals than Survey B.
6. Survey A is better than Survey B since it has a higher percentage.

A group of scientists studied the adhesion of various biofilms to solid surfaces for possible use in environmental technologies. Adhesion assay is conducted by measuring absorbance at A590. If the scientists want the confidence interval to be no wider than 0.55 ${\displaystyle dy\ne -c{m}^2}$ at ${\displaystyle 98\mathrm{\%}}$ confidence interval, how many observations should they take? Assume that the standard deviation is known to be 0.66 ${\displaystyle dy\ne -c{m}^2}$
CHOOSE THE RIGHT ANSWER:
a.39
b.21
c.16
d.25
e.18
f.32

The lengths of the products produced in a production process are controlled. The standard deviation is 0.45 mm. A quality control specialist keeps checking over 40 units randomly every morning. Average length in one day 35.62mm. when it happens. Which of the following is the confidence interval for the mean population mean at ${\displaystyle 95\mathrm{\%}}$ confidence level?
A) ${\displaystyle P\left(60<\mu <70\right)=0,95}$
B) ${\displaystyle P\left(30<\mu <40\right)=0,05}$
C) ${\displaystyle P\left(30<\mu <40\right)=0,95}$
D) ${\displaystyle P\left(60<\mu <70\right)=0,05}$
E) ${\displaystyle P\left(70<\mu <80\right)=0,95}$

margin of error: $130,confidence level: ${\displaystyle 95\mathrm{\%},\sigma =\mathrm{\$}591}$ what is the sample size?

A simple random sample is drawn from a population that is known to be normally distributed. The sample variance, s2, is determined to be 12.6
A. Construct a ${\displaystyle 90\mathrm{\%}}$ confidence interval for ${\displaystyle \sigma }$ if the sample size, n, is 20. (Hint: use the result obtained from part (a)) 7.94 (LB) & 23.66 (UB))
B. Construct a ${\displaystyle 90\mathrm{\%}}$ confidence interval for ${\displaystyle \sigma }$ if the sample size, n, is 30. (Hint: use the result obtained from part (b)) 8.59 (LB) &amp. 20.63 (UB)

The city wishes to estimate the average commute distance for all city employees. They collect a random sample of 97 employees and find a sample mean of ${\displaystyle \bar{x}=7.1\bar{x}=7.1}$ miles. They also assume a population standard deviation of ${\displaystyle \sigma =1.5\sigma =1.5}$ miles. Calculate the z interval to estimate, ${\displaystyle \mu \mu }$, the average commute distance for all city employees with ${\displaystyle 95\mathrm{\%}}$ confidence.
Round your answers to 2 decimal places.
Find the Critical Value z*:
Point Estimate:
Margin of Error:
Write the interval as (lower bound, upper bound)

A study a local high school tried to determine the mean height of females in the US. A study surveyed a random sample of 125 females and found a mean height of 64.5 inches with a standard deviation of 5 inches. Determine a ${\displaystyle 95\mathrm{\%}}$ confidence interval for the mean.
$\begin{array}{|cc|}\hline \text{Confidence Interval}& z\\ 80\mathrm{\%}& 1.282\\ 85\mathrm{\%}& 1.440\\ 90\mathrm{\%}& 1.645\\ 95\mathrm{\%}& 1.960\\ 99\mathrm{\%}& 2.576\\ 99.5\mathrm{\%}& 2.807\\ 99.9\mathrm{\%}& 3.291\\ \hline\end{array}$

A food safety guideline is that the mercury in fish should be below 1 part per million​ (ppm). Listed below are the amounts of mercury​ (ppm) found in tuna sushi sampled at different stores in a major city. Construct a ${\displaystyle 95\mathrm{\%}}$ confidence interval estimate of the mean amount of mercury in the population. Does it appear that there is too much mercury in tuna​ sushi?
$\begin{array}{|ccccccc|}\hline 0.60& 0.82& 0.09& 0.89& 1.29& 0.49& 0.82\\ \hline\end{array}$
What is the confidence interval estimate of the population mean ${\displaystyle \mu ?}$
${\displaystyle ?p\pm <\mu <?p\pm }$

Use the Student's t distribution to find $t_c$ for a 0.95 confidence level when the sample is 24. (Round your answer to three decimal places.)

If ${\displaystyle n=590}$ and ${\displaystyle p=0.93}$, construct a ${\displaystyle 90\mathrm{\%}}$ confidence interval.

Refer to the accompanying data display that results from a sample of airport data speeds in Mbps. Complete parts (a) through (c) below.
a. Express the confidence interval in the format that uses the "less than" symbol. Given that the original listed data use one decimal place, round the confidence intervals limits accordingly.
${\displaystyle 13.05Mbps<\mu <22.15}$ Mbps
b. Identify the best point estimate of ${\displaystyle \mu }$ and the margin of error.
The point estimate of ${\displaystyle \mu }$ is 17.60 Mbps.
The margin of error is ${\displaystyle E=?}$ Mbps.

Assume that we want to construct a confidence interval. Do one of the following, as appropriate: a) find the critical value ${\displaystyle t_{\frac{\alpha }{2}}}$. b) find the critical value ${\displaystyle z_{\frac{\alpha }{2}}}$, or c) state that neither the normal distribution nor the t distribution applies.
Here are summary statistics for randomly selected weights of newborn girls: ${\displaystyle n=249,\bar{x}=31.5hg,s=6.3hg}$. The confidence level is ${\displaystyle 95\mathrm{\%}}$.
A. ${\displaystyle t_{\frac{\alpha }{2}}=?}$
B. ${\displaystyle z_{\frac{\alpha }{2}}=?}$
C. Neither the normal distribution nor the t distribution applies.

As part of an annual review of its accounts, a discount brokerage selects a random sample of 28 customers. Their accounts are reviewed for total account valuation, which showed a mean of $35,100, with a sample standard deviation of $8,800. (Use t Distribution Table.)
What is a ${\displaystyle 98\mathrm{\%}}$ confidence interval for the mean account valuation of the population of customers? (Round your answers to the nearest dollar amount.)
${\displaystyle 98\mathrm{\%}}$ confidence interval for the mean account valuation is between $ and $

How do you solve?
A healthcare provider monitors the number of CAT scans performed each month in each of its clinics. The most recent year of data for a particular clinics follows (the reported variable is the number of CAT scans each month expressed as the number of CAT scans per thousand members of the health plan):
2.31,2.09,2.36,1.95,1.98,2.25,2.16,2.07,1.88,1.94,1.97,2.02.
Find a two-sided ${\displaystyle 95\mathrm{\%}}$ confidence interval for the standard deviation. What should you do to address any reservations about this confidence interval?
Round your answers to two decimal places (e.g. 98.76).
$\le \sigma \le$

Of 1000 randomly selected cases of lung cancer 843 resulted in death within 10 years. Construct a ${\displaystyle 95\mathrm{\%}}$ two-sided confidence interval on the death rate from lung cancer.
a) Construct a ${\displaystyle 95\mathrm{\%}}$ two-sides confidence interval on the death rate from lung cancer. Round your answer 3 decimal places.
${\displaystyle ?\le p\le ?}$
b) Using the point estimate of p obtained from the preliminary sample what sample size is needed to be ${\displaystyle 95\mathrm{\%}}$ confident that the error in estimatimating the true value of p is less than 0.00?
c) How large must the sample if we wish to be at least ${\displaystyle 95\mathrm{\%}}$ confident that the error in estimating p is less than 0.03 regardless of the value of p?

Find ${\displaystyle t_{\frac{\alpha }{2}},n=20}$ for the ${\displaystyle 95\mathrm{\%}}$ confidence interval for the mean.