E370 Exam 3

The purpose of generating a confidence interval for the mean is to provide an estimate for the value of the population mean.

True

In hypothesis testing, the tentative assumption about the population parameter is

the null hypothesis

It is known that the population is not normally distributed. Also, the population standard deviation is not known. A sample of 6 items is selected from this population to develop an interval estimate for the mean of the population (µ). Choose the correct alternative below.

the sample size must be increased for constructing the confidence interval

For the interval estimation of µ when σ is known and the sample is large, the proper distribution to use is

the standard normal distribution

Over the past several years, the proportion of one-person households has been increasing. The Census Bureau would like to test the hypothesis that the proportion of one-person households exceeds 0.27. A random sample of 125 households found that 43 consisted of one person. To conduct the hypothesis test, what distribution would you use to calculate the critical value and the p-value?

the standard normal distribution

When conducting a hypothesis test for the population mean when σ is known and the sample size is 30 or more, the distribution we use for computing the critical value and the p-value is

the standard normal distribution

`Hypothesis testing is used

to determine whether a statement about the value of a population parameter should or should not be rejected

Calculation of a P Value for a 2 tail test

2 * P(z>Zxbar) or 2 * P(Z<Zxbar)

The supervisor of a production line wants to check if the average time to assemble an electronic component is different from 14 minutes. Assume that the population of assembly time is normally distributed with a standard deviation of 3.4 minutes. The supervisor times the assembly of 14 components, and finds that the average time for completion is 11.6 minutes. How would you calculate the p-value for the hypothesis test?

2 × P ( z < − 2.64 )

An accountant claims to be able to complete a standard tax return in under an hour. For a random sample of 24 tax returns, the accountant averaged 63.2 minutes. Assume that the population standard deviation is equal to 7.7 minutes and the population is normally distributed. What is the test statistic for the hypothesis test to check accountant's claim?

2.04

A hypothesis is:

an assumption about value of a population parameter

After computing a confidence interval, the user believes the results are meaningless because the width of the interval is too large. Which one of the following is the best recommendation?

increase the sample size

A confidence interval for the mean

is an interval estimate around a sample mean that provides us with a range within which the true population mean is expected to lie

The confidence interval for the proportion

is an interval estimate around a sample proportion that provides us with a range of where the true population proportion lies

The p-value

is the probability of observing a sample mean at least as extreme as the one selected for the hypothesis test, assuming the null hypothesis is true with equality

H_1: \mu < 40 H 0 : μ ≥ 40 ; H 1 : μ < 40 A sample of 49 observations provides a sample mean of 38 and a sample standard deviation of 7. Compute the value of the test statistic.

-2

Suppose that you want to test: LaTeX: H_0: \mu \geq 0.54; \text{ } H_1: \mu < 0.54 H 0 : μ ≥ 0.54 ; H 1 : μ < 0.54 based on a sample of n = 25 and known population standard deviation of 13.2. What is the appropriate boarder of the rejection region for the test at 0.03 significance level?

-z0.03

If an interval estimate is said to be constructed at the 90% confidence level, the significance level would be

0.1

A simple random sample of 64 observations was taken from a large population. The sample mean and the standard deviation were determined to be 320 and 120 respectively. The estimated standard error of the mean is

15

A simple random sample of 5 observations from a population containing 400 elements was taken, and the following values were obtained: 12 18 19 20 21 A point estimate of the mean is

18

The screening process for detecting a rare disease is not perfect. Researchers have developed a blood test that is considered fairly reliable. It gives a positive reaction in 98% of the people who have a disease. However, it erroneously gives a positive reaction for 3% of the people who do not have the disease. Consider the null hypothesis "the individual does not have the disease". What is the probability of Type I error if the new blood test is used?

3%

In order to determine an interval for the mean of a population with unknown standard deviation, a sample of 61 items is selected. The mean of the sample is determined to be 23. The number of degrees of freedom for the critical t value is

60

All else being equal, a 90% confidence interval will be wider than a 95% confidence interval.

False

A production line operation is designed to fill cartons with laundry detergent to a mean weight of 32 ounces. A sample of cartons is periodically selected and weighted to determine whether overfilling or underfilling is occurring. If the sample data led to a conclusion of underfilling or overfilling, the production line would be shut down and adjusted to obtain a proper filling. The correct hypothesis statement for this test would be

Ho: u = 32 H1: u =/32

The value added and subtracted from a point estimate in order to develop an interval estimate of the population parameter is known as the

Margin of Error

Calculation of P Value for a t lower tail test

P(t<txbar)

Whenever estimating the confidence interval for the population proportion, which of the following is required?

Sample size and populations proportion should be such that NP >= 5 and n(1-p) >=5

Medicare would like to test if the average monthly rate for one-bedroom assisted-living facility is different from $3,300. Suppose that you collect sample information and, based on the hypothesis test, determine that the z test statistic is equal to 1.98 while the critical value of z is 2.05. How would you state the conclusion?

Since LaTeX: -z_{\alpha/2} = -2.05 < z_{\bar x} = 1.98 < z_{\alpha/2} = 2.05 − z α / 2 = − 2.05 < z x ¯ = 1.98 < z α / 2 = 2.05 , do not reject H0. Therefore, there is not enough evidence to conclude that the monthly rate for one-bedroom assisted-living facility is different from $3,300.

The manager of an automobile dealership is trying to understand if a new bonus plan has increased sales. Previously, the mean sales rate per salesperson was five automobiles per month. Suppose that, the manager conducts a hypothesis test and, based on the hypothesis test, she determines that the p-value equals 0.0062. Using the significance level LaTeX: \alpha=0.05 α = 0.05 , how would you state the conclusion for the test?

Since the p-value = 0.0062 < 0.05, reject H0. Therefore, there is an evidence that the new bonus plan is more efficient than the old one.

The error of rejecting a true null hypothesis is

a Type I error

A 95% confidence interval for the population mean is determined to be between 100 and 120. If the confidence level is reduced to 90%, the confidence interval for µ

becomes narrower

In order to construct the interval estimate of the population mean when σ is known and the sample is very small, the population

must have a normal distribution

In the point estimation

data from the sample is used to estimate the population parameter

As the sample size increases, the margin of error

decreases

Suppose that you are working on an upper tail test. As the test statistic becomes larger, the p-value

gets smaller

Given the population standard deviation, the sample size of 59 and the sample mean of 879, you calculate the value of the test statistic LaTeX: z_{\bar x} z x ¯ and determine the p-value = 0.017 for the test. This p-value means that

if the population mean was equal to 800, then there is a 1.7% probability of obtaining the sample mean of 879 or greater in the sample of 59 observations.

A point estimate

is a single value that best describes the population of interest X bar S P bar

A Type II error

occurs when we fail to reject the null hypothesis when it is not true

The sample standard deviation is the point estimator of the

population standard deviation

The level of significance is the

probability of Type 1 error

An interval estimate

provides a range of values that best describes the population

Increasing the sample size while keeping the confidence level constant will

reduce the margin of error

When s is used to estimate σ, the margin of error is computed by using

t distribution

Breyers is a major producer of ice cream and would like to test if the average American consumes more than 17 ounces of ice cream per month. A random sample of 25 Americans was found to consume an average of 19 ounces of ice cream last month. The standard deviation for this sample was 5 ounces. Breyers would like to set LaTeX: \alpha = 0.025 α = 0.025 for the hypothesis test. It is known that LaTeX: z_{\alpha}=1.96 z α = 1.96 and LaTeX: t_{\alpha}=2.06 t α = 2.06 for the df = 24. Also, it is established that the ice cream consumption follows the normal distribution in the population. The conclusion for this hypothesis test would be

t x ¯ < t α . So, we do not reject the null and cannot conclude that the average amount of ice cream consumed per month is greater than 17 ounces.

In hypothesis testing, if the null hypothesis is rejected,

the alternative hypothesis is true

For a one-tail (upper) hypothesis test, if the z-test statistic exceeds the critical value, we do not reject the null hypothesis.

False

Given that a 95% confidence interval is (6.5, 12.5), we can state that there is a 95% probability that the true population mean is between 6.5 and 12.

False

If the hypothesis test is to be used in an attempt to prove a particular point of view, you would assign the null hypothesis to establish the point of view you favor.

False

If we fail to reject H 0 when, in fact, it is not true, we commit Type I error.

False

The confidence interval for the mean is symmetrical around the population mean.

False

The margin of error can be reduced by reducing the size of the sample.

False

The p-value of a test is the probability that the null hypothesis is true.

False

When testing for the population mean when LaTeX: \sigma σ is known and a sample size is greater than 30, the population must be normally distributed in order for the conclusions to be reliable.

False

When the p-value is greater than the significance level, the conclusion for the hypothesis test is to reject the null hypothesis.

False

Last year, a soft drink manufacturer had 21% of the market. In order to increase their portion of the market, the manufacturer has introduced a new flavor in their soft drinks. A sample of 400 individuals participated in the taste test and 100 indicated that they like the taste. We are interested in determining if more than 21% of the population will like the new soft drink at the significance level 0.05.

H0: p <= 0.21 H1: p > 0.21

The National Association of Realtors reported that 26% of home buyers in the state of Florida were foreigners in 2015. In 2017, a group of students from Indiana University conducted a study and, based on a sample of 100 people, concluded that 28% of home buyers in the state of Florida are foreigners. So, the members of the group think that the proportion has increased since then. The correct hypothesis statement for the student's group to test is

H0: p <= 0.26 H1: p > 0.26

Representatives of a large national union announced that the fraction of women in the union was equal to one-half in the previous year. You are interested in testing whether there has been a change in the fraction of women this year. How would you set up the hypothesis test?

H0: p = 0.5 H1: p =/ 0.5

Travelocity would like to test if the average roundtrip airfare between New York and London differs from $1,400. The correct hypothesis statement for this test would be

H0: u=1400 H1: u does not equal 1400

A golfer claims that his average golf score at the course he plays regularly is less than 90. The correct hypothesis statement for this golfer to prove his claim would be

H0: u>= 90 H1: u< 90

Suppose that, instead of the critical value approach, you decided to use the p-value approach and found that the p-value for the test in this problem is equal to 0.0228. This p-value is best interpreted as the following:

If the true proportion of female employees in the company was 50%, then there would be a 2.28% chance of observing a proportion of females equal to or below 45% in the sample of 400 individuals.

Two Tail test conlcusion

Our sample of 49 smartphone users provides sufficient evidence to reject the null hypothesis, so we support the alternative hypothesis that the average data use is not equal to 1.8 gigabytes per month

A professor of statistics wants to test that the average amount of money a typical college student spends per day during spring break is over $70. Based upon previous research, the population standard deviation is estimated to be $17.32. The professor surveys 35 students and finds that the mean spending is $72.43. How would you calculate the p-value for this test?

P(z>0.83)

Calculation of P Value for a z Upper tail test

P(z>zxbar)

One Tail P Value Conclusion

The p-value represents the probability of obtaining a sample mean of 8,120 hours or greater if the true population mean is 8,000 hours

One Tail Hypothesis test Conclusion example

Since the value of the test statistic ̅= 1.44 is smaller than the critical value = 1.645, we can conclude that... According to our sample of 36 new CFL bulbs, there is not enough evidence to support Edalight's claim that the average life of these bulbs exceeds 8,000 hour.

The p-value is best interpreted as the following:

The p-value indicates the probability of observing the proportion of people who like new flavor equal to 25% or greater in the sample of 400 individuals, if actual proportion of people who like the new flavor is 21%.

For a one-tail (upper) test, the p-value is computed as the probability of obtaining a z-test statistic at least as large as that provided by the sample.

True

For a two-tail hypothesis test, if the absolute value of the z-test statistic exceeds the critical value (i.e. exceeds the absolute value of the critical z), we reject the null hypothesis

True

The definition of a 90% confidence interval is that we expect that close to 90% of a large number of sample means drawn from a population will produce confidence intervals that include that population's mean.

True

The hypothesis statement H: μ < 60 is an example of an alternative hypothesis in a one-tail test.

True

The point estimate for the population mean will always be found within the limits of the confidence interval for the mean.

True

There is no guarantee that every confidence interval taken from a population will include the population mean.

True

When testing for the population mean when the population standard deviation is known (and a sample size is greater than 30), a standard normal distribution is used to calculate the critical value and the p-value.

True

When the population standard deviation is unknown, we substitute the sample standard deviation in its place to calculate confidence intervals.

True

When we use the z-distribution to calculate a confidence interval, we need to assume that the population of interest follows the normal probability distribution if the sample size is small (less than 30 observations).

True

Which of the following statements is correct? (this question is a little tricky)

We cannot establish a claim if the null hypothesis is not rejected

Given that a 95% confidence interval for the population mean is (6.5, 12.5), we can state that

We expect that 95% of all possible sample means drawn from the population will produce confidence intervals that include the population's mean (which makes us 95% confident that interval (6.5, 12.5) contains the population mean)

How would you be wrong if the town's population was in fact 1,400 but you computed the confidence interval not taking this into account?

We would fail to take into account the finite population correction factor which would make the confidence interval wider.

Are there any additional assumptions needed in this problem to ensure that computed confidence interval is reliable?

Yes, the population needs to be normally distributed because the sample is too small to apply CLT.

Assuming that the population standard deviation is known and alpha α is the level of significance, the null hypothesis will be rejected if

Z xbar > Z alpha

As the number of degrees of freedom for the t distribution increases, the difference between the t distribution and the standard normal distribution

becomes smaller

For a given confidence level and sample size, which of the following is true in the interval estimation of the population mean when σ is known?

if the population standard deviation is larger, the interval is wider.

If a hypothesis is rejected at 5% level of significance, it (Hint: draw a picture for this question)

may be rejected or not rejected at the 1% level

H 0 : μ ≥ 40 ; H 1 : μ < 40 . A sample of 49 observations provides a sample mean of 38 and a sample standard deviation of 7. The null hypothesis will be rejected if

t xbar < -t alpha

When conducting a hypothesis test for the population mean when σ is unknown and the sample size is 30 or more, the distribution we use for computing the critical value and the p-value is

the Student's t-distribution

The Department of Economic and Community Development (DECD) reported that in 2009 the average number of new jobs created per county was 450 (the only information that you know from DECD). Doing a project in your international economics class, you want to determine whether there has been a decrease in the average number of jobs created, and you collect information about 10 countries. To conduct the hypothesis test, what distribution would you use to calculate the critical value and the p-value?

the Student's t-distribution with 9 degrees of freedom

Confidence intervals of the population mean may be created for the cases when the population standard deviation is known or unknown. How are these two cases treated differently?

use the z distribution when pop standard deviation is known and T distribution when it is not

The confidence intervals for the population proportion are generally based on

z distribution

The Margin of Error

̅is the amount added and subtracted to the point estimate to form the confidence interval

Confidence Intervals for the Mean, σKnown assumptions

•The population standard deviation (σ) is known •Sampling distribution of is normal •The sample size is at least 30 (n≥ 30) •Population distribution is normal