Estimation and Confidence Intervals
A student found that out of 100 cars that passed an intersection 20 were Toyotas. Create a 95% confidence interval for the population proportion of Toyotas. Given z = 1.96.
0.12 to 0. 28
From a random sample of 20 adults, five indicate that they regularly read books. What is the sample proportion?
0.25
If there is no estimate of the population proportion, what value should be used in calculating the sample size for a population proportion?
0.5
Put the following steps in order from first to last to construct a confidence interval to estimate the population mean from a sample.
1. Calculate the sample mean. 2. Find the sample standard deviation. 3. Find the t-value based on the sample size. 4. Calculate the confidence interval.
The Central Limit Theorem tells us that the sample means follow a normal distribution with mean μ and standard deviation (i.e. standard error) of σ/n⎯⎯√n. This lets us use z-values to set confidence intervals. Match the confidence levels to the z-values. 1.) 68% confidence 2.) 95% confidence 3.)99% confidence
1.) z = 1, interval μ±σ/n⎯⎯√n 2.) z = 1.96, interval μ±1.96σ/n⎯⎯√n 3.) z = 2.58, interval μ± 2.58σ/n⎯⎯√
Calculate a 95% confidence interval for a sample mean of 20 with a sample standard deviation of 10 and a sample size of 9. The answer should be accurate to the nearest decimal. Given t = 2.306.
12.31 to 27.69
Suppose you want to estimate the population mean with 99% confidence with a margin of error of 2.5. If the estimate of the population standard deviation is 11, what sample size is required?
129
Calculate a 90% confidence interval for a sample mean of 15 with a sample standard deviation of 5 and a sample size of 25. The answer should be accurate to the nearest decimal. Given t = 1.711.
13.29 to 16.71
A sample of size 16 is taken from a population with standard deviation 8. The sample mean is 22. Which of the following is a 95% confidence interval for the population mean?
18.08 to 25.92
What z-value is used to construct a 98% confidence interval for the population mean when the population standard deviation is known?
2.33
A large jar contains a mixture of white and black beans. In a randomly chosen handful of 120 beans 40 were black. Create a 98% confidence interval for the proportion of black beans in the jar. Express your answer to the nearest percent.
23% to 43%
A large jar contains a mixture of white and black beans. A small sample of beans was found to be 1/4 black beans. What size sample would be needed to estimate the proportion of black with an error of no more than 0.05 and a confidence level of 95%?
289
A company has a total of 100 employees. From a random sample of 33 employees, the average age is found to be 44 years with a standard deviation of 3 years. Construct a 99% confidence interval to estimate the population mean age.
42.8 to 45.2
A sample of size 25 is taken from a population with standard deviation 6. The sample mean is 10. Which of the following is a 99% confidence interval for the population mean?
6.9 to 13.1
Suppose you would like to estimate what proportion of students work full-time. If you have no estimate of the population proportion, what sample size would be needed to estimate the proportion with an error of no more than 0.04 and a confidence level of 95%?
601
Suppose you want to estimate the population mean with 95% confidence with a margin of error of 2. If the estimate of the population standard deviation is 8, what sample size is required?
62
Suppose that, after using the sample size formula, you find n=83.25. What sample size should you use?
84
A restaurant had 80 customers yesterday. From a random sample of 25 customer bills, the average bill is found to be $18 with a standard deviation of $4. Construct a 95% confidence interval to estimate the population mean bill.
$16.62 to $19.38
Which of the following formulas would you use to calculate the sample size n for a mean?
(zσ/E)^2
A proportion would be especially useful in which one of the following cases?
Estimating the percentage of students who have full-time jobs.
In constructing a confidence interval to estimate the population mean from a sample, which of the following steps are necessary only when σ is not known? Select all that apply.
Find the t-value for the proper distribution, based on the sample size. Find the sample standard deviation.
Why is it necessary to apply the finite population correction factor when a sample is a significant part of the population?
If a sample is a larger part of the population, it will give a better estimate.
Which of the following statements correctly describe the role of the population standard deviation, σ, in creating a confidence interval for the population mean?
If σ is not known, we estimate it using the sample standard deviation, s. If σ is known, we use it to calculate the confidence interval.
The binomial conditions must be met before we can develop a confidence interval for a population proportion. Which two of the following are binomial conditions?
We can define two outcomes, success and failure. The probability of success is the same for all trials.
Which of the following must be true so that the standard normal distribution can be used to construct a confidence interval for the population proportion?
nπ≥5 and n(1-π)≥5
Which of the following is the formula used to calculate a confidence interval for the population proportion?
p±z⋅√p(1−p)/n
In creating a confidence interval for the population mean, if σ is unknown, we estimate it using which of the following?
s
When the sample is a significant part of a finite population, we need to adjust the standard error, and thus the confidence interval by using the formula XX± ts√nsn(√N−nN−1)N-nN-1. Match the variables to their definition. t s n N √N−n/N−1
the Student's t cutoff value sample standard deviation sample size population size Finite population correction factor
A confidence interval for the population proportion is calculated using the formula p ± z√p(1−p)np(1-p)n. Match the variables to their description. p z n π
the sample proportion the confidence level the sample size the population proportion that is being estimated
What doe the symbols in the formula p = xnxn stand for? Match the variables to their description. p x n
the sample proprtion "successes" in the sample sample size
Which of the following is the best point estimate of the population mean, μ?
x bar
When the population standard deviation is known, the confidence interval for the population mean is based on the:
z-statistic
Which of the following is the correct formula for finite-population correction factor, FPC =?
√N−n/N−1
Which of the following considerations require a larger sample size? Select all that apply.
A higher level of confidence. A smaller margin on error.
What is a Confidence Interval? Choose the best description.
A range of values, created using a sample, within which a population parameter has a certain probability of occurring.
Which of the following statements is the best definition of a point estimate?
A sample statistic that estimates a population parameter.
When we speak of an "80% confidence level," what are we referring to?
An 80% probability that a parameter will be within a specified interval.
What is the effect of the finite population correction factor on a sample based estimate?
It makes the estimate more precise (narrower) for larger samples.
Identify which of the following are traits that apply to the meaning of "proportion." Select all that apply.
It refers to a fraction, ratio, or percent. It can refer to either a sample or a population.
Why is it important to know the population standard deviation when estimating the population mean?
Knowing σ lets us use the standard normal distribution to construct a confidence interval.
How does sample size affect the width of the confidence interval for the population mean?
Larger sample sizes result in narrower intervals.
How does the standard deviation of the population affect the width of the confidence interval for the population mean?
Larger values of the population standard deviation result in wider intervals.
What should you do if your calculation for appropriate sample size gives a fractional number?
Round up to the next whole number.
For a given confidence level, how does a confidence interval calculated using the t-value compare to one calculated using the z-value?
The confidence interval from the t-value is wider.
What is the difference between the a confidence interval and the level of confidence?
The confidence interval is a range of values, the level of confidence is the probability for that range of values.
Which of the following items are valid considerations in the choice of sample size? Select all that apply.
The desired level of confidence. The population dispersion. The margin of error the researcher will tolerate.
In choosing a sample size for a study, we need to know the population standard deviation. Which of the following could be used to estimate the population standard deviation? Select all that apply.
The population standard deviation from a comparable study. The sample standard deviation from a pilot study.
What is meant by the term "level of confidence"?
The probability that a parameter will be found within the confidence interval.
A confidence interval is constructed using the formula X± z σn√σn. Match the symbols to their definition. 1.) xbar 2.) σ 3.) n 4.) σ/√nσn 5. z
The sample mean The population standard deviation The sample size The standard error. Standard normal confidence level.
A confidence interval is constructed using the formula X⎯⎯⎯±tsn√X¯±tsn. Drag and drop the definitions against the corresponding symbols. Instructions Xbar s n s√n
The sample mean The sample standard deviation The same size The standard error
Knowing the population standard deviation allows us to choose an appropriate sample size for a study. Which two of the following could be used to estimate the population standard deviation?
The sample standard deviation from a pilot study. One sixth of the population range.
The formula for estimating the sample size for a study is n = (zσ/E)^2. Match the variables to their description. n z σ E
The size of the sample The standard normal value for the chosen confidence level. The population standard deviation. The maximum allowable error.
Which of the factors listed below determine the width of a confidence interval? Select all that apply.
The size of the standard error. The chosen level of confidence.
For a specific confidence level, how do the t-distribution and the z-distribution compare?
The t-distribution is more spread out than the z-distribution.
It refers to a fraction, ratio, or percent. It can refer to either a sample or a population.
The t-distribution is more spread out than the z-distribution.
When we want to estimate the mean of a population using a sample, but do not know the population standard deviation, which of the following steps are required? Select all that apply.
Use the sample standard deviation as an estimate of the population standard deviation. Use the t-distribution instead of the Standard Normal to find the confidence interval.
If we are estimating the population mean using a sample, under what circumstances would we use the t-distribution?
When we don't know the population standard deviation.
Choose the correct formula for calculating the confidence interval for the mean when the population standard deviation is not known.
Xbar ± t s/√n
Which of the following formulas would you use to construct a confidence interval for the mean with σ known?
Xbar ± z σ/n√
Which of the following is the correct formula for choosing the sample size n for an estimate of population proportion?
n = π(1 = π)(z/E)^2
