MATH 1680 Chapter 9

When constructing​ 95% confidence intervals for the mean when the parent population is right skewed and the sample size is​ small, the proportion of intervals that include the population mean approaches _______ as the sample​ size, n, increases.

0.95

As the sample size, n, increases, what happens to the margin of error?

As the sample size, n, increases, the margin of error decreases. Therefore, larger sample sizes will result in narrower confidence intervals.

In a survey conducted by the Gallup​ Organization, 1100 adult Americans were asked how many hours they worked in the previous week. Based on the​ results, a​ 95% confidence interval for the mean number of hours worked had a lower bound of 42.7 and an upper bound of 44.5. Provide two recommendations for decreasing the margin of error of the interval.

Decrease the confidence level and increase the sample size

As the level of confidence increases, what happens to the critical value?

It increases

What does it mean when we say that the procedure for constructing a confidence interval is robust?

Notice that a confidence interval about μ can be computed for non-normal populations even though Student's t-distribution requires a normal population. This is because the procedure for constructing the confidence interval is robust—it is accurate despite minor departures from normality

What type of data are needed to construct a confidence interval for a population proportion, p?

Qualitative with 2 outcomes

What type of data are needed to construct a confidence interval for a population mean, ?

Quantitative

What was the name of the brewery that Gosset worked for? What pseudonym did he choose to publish his results about a model that accounts for the additional variability introduced by using s in place of when determining margin of error?

The Guinness Brewery. Chose Student as his pseudonym

What does the level of confidence represent?

The level of confidence represents the expected proportion of intervals that will contain the parameter if a large number of different samples is obtained. The level of confidence is denoted (1- a)100%

Any sample statistic that is in the tails of the sampling distribution will result in a confidence interval that does not include the population parameter.

True

As the level of confidence increases, the size of the interval increases

True

As the sample size increases, the margin of error decreases

True

If the sample size is quadrupled, the margin of error will be cut in half.

True

If the normality condition is not satisfied, how does the proportion of intervals that capture the parameter compare to the level of confidence?

When the normality condition is not satisfied, the proportion of intervals that capture the parameter is below the level of confidence.

Put the following in order for the most area in the tails of the distribution. ​(a) Standard Normal Distribution ​(b) Student's​ t-Distribution with 15 degrees of freedom. ​(c) Student's​ t-Distribution with 30 degrees of freedom.

b, c, a

When constructing​ 95% confidence intervals for the mean when the parent population is right skewed and the sample size is​ small, the proportion of intervals that include the population mean is _______ 0.95.

below

The t-distribution gives a ________ critical value than the z-distribution, so the width of the confidence interval is wider when it is constructed using Student's t-distribution.

larger

If the normality requirement is not satisfied​ (that is, ​np(1−​p) is not at least​ 10), then a​ 95% confidence interval about the population proportion will include the population proportion in​ ________ 95% of the intervals.

less than

The​ _______ represents the expected proportion of intervals that will contain the parameter if a large number of different samples of size n is obtained. It is denoted​ _______.

level of confidence; (1-a)100%

Besides the facts that the sample must be obtained by simple random sampling or through a randomized experiment and that the sample size must be small relative to the size of the population, what other condition must be satisfied?

n > 30 (good to go) n < 30 we create a box plot/

Besides the fact that the sample must be obtained by simple random sampling or through a randomized experiment, list the two conditions that must be met when constructing a confidence interval for a population proportion, p.

np^ (1-p^) > 10 and n<0.05N

What is the form of confidence interval estimates for a population parameter?

point estimate + or - margin of error

Whether a confidence interval contains the population parameter depends solely on the value of ______________________

the sample statistic.

Any sample proportion that is more than 1.96 standard errors from the population proportion ______ result in a confidence interval that does not contain p. This will happen in 5% of all possible samples (those sample proportions in the tails of the distribution).

will

Constructing a Confidence Interval for a Population Proportion using StatCrunch

- Stat > proportion stats > one sample > with summary -Enter number of successes (x) and number of observations (n) - Choose the confidence interval radio button, enter the level of confidence (leave method as Standard-Wald) - Click compute

As the number of samples​ increases, the proportion of​ 95% confidence intervals that include the population proportion approaches:

0.95

State six properties of the t-distribution.

1. The t distribution is different for different degrees of freedom 2. The t distribution is centered at 0 and is symmetric about 0 3. The area under the curve is 1. The area under the curve to the right of 0 equals the area under the curve to the left of 0 which equals 1/2 4. As t increases or decreases without bound, the graph approaches, but never equals 0 5. The area in the tails of the t-distribution is a little greater than the area in the tails of the standard normal distribution, because we are using s as an estimate of sigma, thereby introducing further variability into the t-statistic 6. As the sample size n increases, the density curve of t gets closer to the standard normal density curve. This result occurs because, as the sample size increases, the values of s get closer to the value of sigma, by the Law of Large Numbers

List the three conditions required for constructing a confidence interval for a population mean μ

1. sample data come from a simple random sample or randomized experiment 2. sample size is small relative to the population size (n < 0.05N) 3. the data come from a population that is normally distributed with no outliers or the sample size is large

f a​ 95% confidence interval results in a sample proportion that does not include the population​ proportion, then the sample proportion is more than _____ standard errors from the population proportion.

1.96

What percent of sample proportions results in a 98​% confidence interval that does not include the population​ proportion?

2%

What percent of sample proportions results in a 98​% confidence interval that includes the population​ proportion?

98%

State the interpretation of a confidence interval.

A (1−α)⋅100% confidence interval indicates that (1−α)⋅100% of all simple random samples of size n from the population whose parameter is unknown will result in an interval that contains the parameter.

Give the definition for a confidence interval for an unknown parameter.

A confidence interval for an unknown parameter consists of an interval of numbers based on a point estimate.

State the definition of a point estimate.

A point estimate is the value of a statistic that estimates the value of a parameter.

List the critical value associated with the given level of confidence.

A) 90% : 1.645 B) 95% : 1.96 C) 99% : 2.575

Because the sample mean and sample standard deviation are not resistant to outliers, sample data should always be inspected for serious departures from normality and for outliers. What tools can be used to check for serious departures from normality and for outliers?

If a data set has outliers, the confidence interval is not accurate because neither the sample mean nor the sample standard deviation is resistant to outliers. Sample data should always be inspected for serious departures from normality and for outliers. This is easily done with normal probability plots and boxplots.

The 95% in a 95% confidence interval represents the proportion of all samples that will result in intervals that include the population proportion.

In practice, we construct only one confidence interval based on one sample. We do not know whether the sample results in a confidence interval that includes the parameter, but we do know that if we construct a 95% confidence interval, it will include the unknown parameter 95% of the time.

What is the point estimate for a population mean mu?

The sample mean x-

What does tα represent?

The t value whose area under the t-distribution to the right of tα is (α) alpha. The shape of the t-distribution depends on the sample size, n. Therefore, the value of tα depends not only on α, but also on the degrees of freedom, n−1. In Table VII, the far left column gives the degrees of freedom (df). The top row represents the area under the t-distribution to the right of some t-value

What happens to the proportion of intervals that capture the population mean as the sample size increases?

They increase too.

Explain why the​ t-distribution has less spread as the number of degrees of freedom increases.

The​ t-distribution has less spread as the degrees of freedom increase​ because, as n​ increases, s becomes closer to σ by the law of large numbers.

For a 95% confidence interval, any sample proportion that lies within 1.96 standard errors of the population proportion will result in a confidence interval that includes p.

This will happen in 95% of all possible samples.