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Chapter 10 Review Objectives

10.1 Objectives:Ø Understand the definitions of the null and the alternative hypothesis and their roles in the hypothesis testing procedure.Ø Be able to construct and write null and appropriate null hypothesis when given problems in context.

10.2 Objectives:Ø Understand the definitions Type I and Type II error and the role these concepts play in hypothesis testing.Ø Be able to identify and discuss the relevance of both types of error in problem contexts.Ø Have a sense that error is inevitable because we are sampling, but the error can be quantified and managed.

10.3 Objectives:Ø Understand the definition of a test statistic and its role in hypothesis testingØ Understand the definition of P-value and its role in hypothesis testingØ Be able to use the hypothesis testing procedure to test a hypothesis about a population proportion

10.3 Objectives:Ø Have a sense that the hypothesis testing procedure for testing a hypothesis about a population proportion is based on the sampling distribution of the sample proportion, including relevant assumptions

Summary of the Large-Sample z Test for p

Null hypothesis: H0: p = hypothesized valueTest Statistic:

Alternative Hypothesis: P-value:Ha: p > hypothesized value Area to the right of calculated zHa: p < hypothesized value Area to the left of calculated zHa: p ≠ hypothesized value 2(Area to the right of z) of +z

or 2(Area to the left of z) of -z

nPpppz

)1(ˆ

−−

=

Summary of the Large-Sample z Test for p Continued . . .

Assumptions:

1. p is a sample proportion from a random sample

2. The sample size n is large. (np > 10 and n(1 - p) > 10)

3. If sampling is without replacement, the sample size is no more than 10% of the population size

10.4 Objectives:• Be able to use the hypothesis testing procedure to test hypotheses about a population mean• Have a sense that the hypothesis testing procedure for testing hypotheses about a population mean is based on the sampling distribution of the sample mean, including the relative assumptions.

Let’s review the assumptions for a confidence interval for a population mean

2) the sample size n is large (n > 30), and3) σ, the population standard deviation, is known

or unknown

1) x is the sample mean from a random sample,

n

xz σµ−=

This is the test statistic when σ is known.

nsxt µ−=

This is the test statistic when σ is unknown.

P-value is area under the z curve

P-value is area under the t curve with df=n-1

The assumptions are the same for a large-sample hypothesis test for a

population mean.

The One-Sample t-test for a Population Mean

Null hypothesis: H0: µ = hypothesized valueTest Statistic:

Alternative Hypothesis: P-value:Ha: µ > hypothesized value Area to the right of calculated t

with df = n-1Ha: µ < hypothesized value Area to the left of calculated t

with df = n-1Ha: µ ≠ hypothesized value 2(Area to the right of t) of +t

or 2(Area to the left of t) of -t

nsxt µ−=

The One-Sample t-test for a Population Mean Continued . . .

Assumptions:

1. x and s are the sample mean and sample standard deviation from a random sample

2. The sample size n is large (n > 30) or the population distribution is at least approximately normal.

10.5 Objectives:• Understand the effects of sample size, significance level, and the value of the parameter on the power of a hypothesis test.• Have a sense on how power is calculated for tests based on the standard normal or t distributions.

Chapter 10 Review

Review Questions

1. When performing hypothesis tests, there are assumptions that must be met in order for the test to be appropriate. Describe how you would check these assumptions for a hypothesis test about a population mean when the raw data from a small sample is available.1) The data must be from a random sample from the population of interest

2) The sample size must be large (at least 30) OR the population must be approximately normal. To check for population normality, the sample should be graphed to see if it is plausible that it came from a normally distributed population.

2. When performing tests of hypotheses, there are assumptions that must be met in order for the test to be appropriate. For the test of a hypothesis about a population proportion, describe how you would check the assumptions.There are 2 conditions which need to be met:

1) The data needs to be from a random sample from the population of interest

2) The sample size needs to be large. To check this, see if np >10 and n(1- p) >10 .

3. It is well known that higher vertebrates –mammals and birds – exhibit lateralized behaviors; in humans this is referred to as “handedness.” An investigator recently observed the coiling behavior of cottonmouth snakes. He created a “laterality index” that measured the tendency for snakes to coil clockwise or counterclockwise. If the snakes failed to exhibit laterality they would have a laterality index equal to 0.5. The investigator wishes to determine whether juvenile cottonmouths exhibit handedness

a) What is the appropriate null hypothesis in this study?

Ho: µ = 0.5

b) What is the appropriate alternative hypothesis in this study?

Ha: µ ≠ 0.5

c) In the context of this study, describe a Type I and a Type II error

A Type I error is to decide that the average laterality index is different from 50 if the population mean laterality was in fact 50.

A Type II error is to fail to decide that the average laterality index is different from 50 if the population mean laterality was in fact different from 50.

4. In cities and towns on the borders between states there is “flight” across state lines to avoid high state taxes on gasoline. Some states have large rivers for borders and tolls to cross bridges. Do these tolls impede traffic to other states to buy cheaper gasoline? To test this hypothesis, an experimental Toll-Free Week will be instituted at the Farmington Bridge in Iowa, where currently approximately 50 cars per day drive back and forth. Let µ denote the true average number of border crossings per day at Farmington if there were no toll.

a) What is the appropriate null hypothesis in this study?

Ho: µ = 50

b) What is the appropriate alternative hypothesis in this study?

Ha: µ > 50

c) In the context of this study, describe a Type I and a Type II error

A Type I error is to decide that the average number of crossings would be greater than 50 if there were no toll, when in fact the average number is still 50.

A Type II error is to decide that the average number of crossings would stay the same (50) if there were no toll, when in fact the average number of crossings would increase.

5. Children develop "representational insight," a connection between an object and a symbol for that object. A random sample of 2-year olds were shown a video of someone putting a toy under 1 of 4 randomly placed boxes in a room familiar to the child. Then they were taken to the room, and asked to "find the toy." The investigators reasoned that a child with representational insight would pick the correct box on the first try. If not, they would find the toy on the first try only 25% of the time. Thirty out of 57 children found the toy by turning over the correct box on the first try.

Do these results provide convincing evidence that the proportion of 2-year old children who choose the correct box on the first try is greater than 0.25? Use a significance level of α =.05 to test the appropriate hypothesis.

6. A boat manufacturer claims that a particular boat and motor combination will burn less than 4.0 gallons of fuel per hour. Fuel consumption for a random sample of 10 similar boats resulted in the data below:

4.06, 4.29, 4.26, 4.64, 4.23, 3.93, 3.64, 4.13, 3.93, 3.86

Is there sufficient evidence to conclude that the manufacturer's claim is correct? Use α=.05 and test the appropriate hypothesis

7. A company provides portable walkie-talkies to construction crews. Their batteries last, on average, 55 hours of continuous use. The purchasing manager receives a brochure advertising a new brand of batteries with a lower price, but suspects that the lifetime of the batteries may be shorter than the brand currently in use. To test this, 8 randomly selected new brand batteries are installed in the same model radio. Here are the results for the lifetime of the batteries (in hours):

45 52 56 55 51 57 48 52

Is there sufficient evidence to conclude that the purchasing manager is correct in his conjecture that the new brand has a shorter average lifetime?