12.1

Hypothesis Test with a Known Standard Deviation

After many years of commuting, Suzy knows that the can bike to her office in 17 minutes with a standard deviation of 1 minute. While shopping, she finds an all-natural energy drink and decides to use it to shorten her commute time. Suzy records 12 morning commutes with an average time of 16.5 minutes in which she drank the energy drink before biking. Is there sufficient evidence to show that the energy drink helped? Conduct a hypothesis test at the 5% significance level.

H0:17H0:17

Ha:<17Ha:<17

Useful tools:

Normal Distribution Calculator

t-Distribution Calculator

1. Conduct a hypothesis test and report the p-value.
2. p=p= ______ (Include four decimal places)
3. At the 5% significance level, we conclude that Suzy bikes faster after drinking the energy drink.

• True
• False

12.2

H0:=24.6H0:=24.6

H1:24.6H1:24.6

Your sample consists of 34 values, with a sample mean of 24.3. Suppose the population standard deviation is known to be 2.13.

a) Calculate the value of the test statistic, rounded to 2 decimal places.

z=z=

b) At =0.025=0.025, the rejection region is

• z>1.96z>1.96
• z>2.24z>2.24
• z<-1.96z<-1.96
• z<-2.24orz>2.24z<-2.24orz>2.24
• z<-2.24z<-2.24
• z<-1.96orz>1.96z<-1.96orz>1.96

c) The decision is to

• Reject the null hypothesis
• Accept the null hypothesis
• Fail to reject the null hypothesis
• Accept the alternative hypotheis

d) Suppose you mistakenly failed to reject the null hypothesis in this problem, what type of error is that?

• Type I
• Type II

12.3

A mayor running for re-election claims that during his term, average municipal taxes have fallen by $250. A conscientious statistician wants to test this claim. He surveys 44 of his neighbors and finds that their taxes decreased (in dollars) as follows:

170, 285, 299, 258, 244, 301, 235, 270, 242, 227, 259, 228, 299, 221, 273, 266, 242, 227, 284, 237, 191, 201, 216, 284, 270, 263, 323, 220, 279, 255, 312, 256, 240, 341, 252, 235, 307, 236, 268, 230, 241, 298, 196, 177

The statistician assumes a population standard deviation of $31. Do you think the statistician should reject the mayor's claim? Why or why not?

Step 1: State the hypothesis.

?

p

=

Step 2: Determine the Features of the Distribution of Point Estimates Using the Central Limit Theorem.

By the Central Limit Theorem, we know that the point estimates are Select an answer

normally distributed

t-distributed

with distribution mean and distribution standard deviation .

Step 3: Assuming the Claim is True, Find the Probability of Obtaining the Point Estimate.

P(P(?

p'

x

?

)=P()=P(?

z

t

?

)=)=

Step 4: Make a Conclusion About the Claim.

What do you think? Based on the probability you calculated in step 3 of obtaining the point estimate, would you reject the claim? Think about your answer to this step yourself; this step is not graded.

12.4

The average McDonald's restaurant generates $2.9 million in sales each year with a standard deviation of 0.8. Darius wants to know if the average sales generated by McDonald's restaurants in Kansas is different than the worldwide average. He surveys 26 restaurants in Kansas and finds the following data (in millions of dollars):

3.6, 2.2, 2.9, 3.7, 3.3, 2.2, 2.3, 2.5, 3, 2.9, 2.5, 2.9, 3.1, 2, 2.7, 3, 3, 2.8, 1.7, 3.3, 3.1, 2.4, 3, 3.6, 2.4, 1.3

Perform a hypothesis test using a 5% level of significance.

Step 1: State the null and alternative hypotheses.

H0:H0: ?

p

?

<

>

=

Ha:Ha: ?

p

?

<

>

=

(So we will be performing a Select an answer

left-tailed

right-tailed

two-tailed

test.)

Step 2: Assuming the null hypothesis is true, determine the features of the distribution of point estimates using the Central Limit Theorem.

By the Central Limit Theorem, we know that the point estimates are Select an answer

normally distributed

t-distributed

with distribution mean and distribution standard deviation .

Step 3: Find the pp-value of the point estimate.

P(P(?

p'

x

?

)=P()=P(?

z

t

?

)=)=

pp-value==

Step 4: Make a Conclusion About the null hypothesis.

Since the pp-value== ?

<

==, we Select an answer

reject

do not reject

the null hypothesis.

• We cannot conclude that the mean sales of McDonald's restaurants in Kansas differ from average McDonald's sales worldwide.
• We conclude that the mean sales of McDonald's restaurants in Kansas differ from average McDonald's sales worldwide.

12.5

The mean age when smokers first start is 13 years old with a population standard deviation of 1.8 years. A researcher thinks that smoking age has significantly changed since the invention of ENDSelectronic nicotine delivery systems. A survey of smokers of this generation was done to see if the mean age has changed. The sample of 30 smokers found that their mean starting age was 12.2 years old. Do the data support the claim at the 5% significance level?

What are the correct hypotheses?

H0: Select an answer

p

s

s

p

x

?

=

>

<

years

H1: Select an answer

p

p

x

s

s

?

>

=

<

years

Based on the hypotheses, find the following:

Test Statistic z = (Give answer to at least 4 decimal places)

Critical Values = (Give answer to at least 4 decimal places)

Based on the above we choose to Select an answer

Accept the null hypothesis

Reject the null hypothesis

Fail to reject the null hypothesis

The correct summary would be: Select an answer

There is enough evidence to reject the claim

There is not enough evidence to reject the claim

There is enough evidence to support the claim

There is not enough evidence to support the claim

that the claim that the mean age smokers first start is different than 13.