Assume that a simple random sample has been selected from a normally distributed
ID: 3133747 • Letter: A
Question
Assume that a simple random sample has been selected from a normally distributed population and test the given claim. Identify the null and alternative hypotheses, test statistic, P-value, and state the final conclusion that addresses the original claim. A simple random sample of 25 filtered 100 mm cigarettes is obtained, and the tar content of each cigarette is measured. The sample has a mean of 19.2mg and a standard deviation of 3.48 mg. Use a 0.05 significance level to test the claim that the mean tar content of filtered 100 mm cigarettes is less than 21.1 mg, which is the mean for unfiltered king size cigarettes. What do the results suggest, if anything, about the effectiveness of the filters?
:
muequals=21.121.1
mg
Upper H 1H1:
mugreater than or equals21.121.1
mg
B.
Upper H 0H0:
muequals=21.121.1
mg
Upper H 1H1:
muless than<21.121.1
mg
C.
Upper H 0H0:
mugreater than>21.121.1
mg
Upper H 1H1:
muless than<21.121.1
mg
D.
Upper H 0H0:
muless than<21.121.1
mg
Upper H 1H1:
mugreater than or equals21.121.1
mg
Identify the test statistic.
tequals=nothing
(Round to three decimal places as needed.)
Identify the P-value.
The P-value is
nothing.
(Round to four decimal places as needed.)
State the final conclusion that addresses the original claim. Choose the correct answer below.
A.
RejectReject
Upper H 0H0.
There is
insufficientinsufficient
evidence to support the claim that the mean tar content of filtered 100 mm cigarettes is less than
21.121.1
mg.
B.
Fail to rejectFail to reject
Upper H 0H0.
There is
sufficientsufficient
evidence to support the claim that the mean tar content of filtered 100 mm cigarettes is less than
21.121.1
mg.
C.
RejectReject
Upper H 0H0.
There is
sufficientsufficient
evidence to support the claim that the mean tar content of filtered 100 mm cigarettes is less than
21.121.1
mg.
D.
Fail to rejectFail to reject
Upper H 0H0.
There is
insufficientinsufficient
evidence to support the claim that the mean tar content of filtered 100 mm cigarettes is less than
21.121.1
mg.
What do the results suggest, if anything, about the effectiveness of the filters?
A.
The results suggest that the filtered cigarettes have the same tar content as unfiltered king size cigarettes.
B.
The results suggest that the filters increase the tar content.
C.
The results
suggestsuggest
that the filters are effective.
D.
The results
do not suggestdo not suggest
that the filters are effective.
E.
The results are inconclusive because the sample size is less than 30.
Explanation / Answer
Assume that a simple random sample has been selected from a normally distributed population.
A simple random sample of 25 filtered 100 mm cigarettes is obtained, and the tar content of each cigarette is measured. The sample has a mean of 19.2mg and a standard deviation of 3.48 mg. Use a 0.05 significance level to test the claim that the mean tar content of filtered 100 mm cigarettes is less than 21.1 mg.
hence sample size=n=25 sample mean=Xbar=19.2 sample standard deviation=s=3.48
level of significance=alpha=0.05
.as the claim is that the mean tar content of filtered 100 mm cigarettes is less than 21.1 mg
the null hypothesis is H0:mu=21.1 mg and the alternative hypothesis is H1: mu<21.1 mg [option B]
now the test statistic is given as
T=(Xbar-21.1)*sqrt(n)/s which under H0 follows a t distribution with degrees of freedom n-1
hence the value of test statistic is t=(19.2-21.1)*sqrt(25)/3.48=-2.729885=-2.730 [answer]
since the alternative hypothesis is left sided hence the p value of the test is
P[T<-2.730] where T follows a t distribution with degrees of freedom=n-1=25-1=24
using MINITAB p=P[T<-2.730]=0.0058369=0.006 [rounded to 3 decimal] [answer]
hence pvalue=0.006<0.05=level of significance
hence the conclusion is
reject H0. there is sufficient evidence to support the claim that the mean tar content of filtered 100 mm cigarettes is less than 21.1 mg [answer]
hence the mean tar content of filtered 100 mm cigarettes is less than that for unfiltered king size cigarettes.
hence the results suggest that the filters are effective because it decreases the mean tar content [answer]
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