math1401-sect09-test3-spring2023 Angie (1)
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Georgia Gwinnett College *
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1401
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Mathematics
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Feb 20, 2024
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MATH 1401 Test 3, Spring 2023 (Section 09)
Name: Angie Sugey Castillo
Complete this document and post to the Assignment Box by Sunday, April 23, 11:59pm
. Please remember the honor code while taking the exam (i.e., do not receive outside help on the exam). If you use a calculator, you must show what you placed in your calculator in order to receive credit.
In the year 2033, Angie Castillo is a leading nurse practitioner. Angie is interested in reducing the mean recovery time for patients after experiencing a serious injury
(assume recovery times are normally distributed). Suppose the mean recovery time is presently 8.6 months. Angie takes a random sample of 46 patients that have experienced serious injury to participate in a new treatment program and finds the sample mean is 8.1 months and a sample standard deviation of 1.2 months. Using α = 0.05, answer the following questions.
a) What is the setup for your null and alternative hypothesis?
H0:µ=8.6 Ha:µ<8.6 (1-sided left)
b) What is the value of the test statistic?
Z*=(8.1-8.6)/(1.2/sqrt(46))= -2.825970826 ~ -2.83 c) What is/are the critical value(s)?
Ha:< and a=0.05 df=n-1 (46-1=45df)
T(45df)=?0.05 area in the left TI-Calculator: 2
nd
vars
invNorm(0.05,0,1,LEFT)= -1.65 Excel: Norm.S.Inv:Probability(0.05)= -1.65 Critical Region:t(45)< -1.65
d) Using the appropriate level of confidence, what is the confidence interval?
Sample Mean+-(Critical Value)(S/sqrt(n)) 8.1+-(1.65)(1.2/sqrt(46)) E= 1.65(1.2/sqrt(46))= 0.2919350732 ~ 0.3 Level of Confidence= Area on t(45) between -1.64(0.05 area to the left) and 1.64(0.05 area to the right)= 1-(0.05+0.05)=0.9 (90%).
8.1+-0.3 Lower (L)= 8.1-0.3=7.8 Upper (U)= 8.1+0.3=8.4 e) What is the interpretation (not your conclusions) of the confidence interval (in the context of the problem) you obtained in part c?
90% confident the recovery time for patients after experiencing a serious injury to participate in a new treatment program lasts between 7.8 months to 8.4 months.
f) What is the p value?
Ha: < and t*(45)=-2.83 p=P(t(45)< -2.83)= TI-Calculator: tcdf(-1E99,-2.83,45)= 0.0034658961 ~ 0.0035
Excel:T.DIST(-2.83,45,TRUE)= 0.003466
~ 0.0035
T*(45) and p TI-Calculator: Stat
Tests
T-Test g) What is the interpretation (not your conclusions) of the p value in the context of the problem you found in part e?
There is 0.35% probability that the sample mean we observed of 8.1 (vs. 8.6) happened by chance.
(7.8,8.4)
T-Test: µ< 8.6 t=-2.825970826 p=0.0035030265
_
X= 8.1
Sx=1.2 n=46
h) What are your conclusions in the context of the problem? Relate your conclusions to the
test statistic and critical value, confidence interval and p value.
Classical Approach: T*(45)= -2.83 and Critical Region: t(45)< -1.65 T*(45)= -2.83< ? -1.65 Yes. Reject H0. Sufficient evidence to conclude a mean reduction in recovery time for patients after experiencing a serious injury to participate in a new treatment program lasts is less than the value
of 8.6 months.
Probability Approach: p= 0.0035 <? a=0.05 Yes. Reject H0. Sufficient evidence to conclude a mean reduction in recovery time for patients after experiencing a serious injury to participate in a new treatment program lasts is less than the value of 8.6 months.
Confidence Interval: 90% CI: (7.8,8.4) Does this interval include the null value 8.6? Yes. Reject H0.
Sufficient evidence to conclude a mean reduction in recovery time for patients after experiencing a serious injury to participate in a new treatment program lasts is less than the value of 8.6 months.
Bonus question (1 point)
What is the title of the classic science fiction novel by Frank Herbert that includes the line “Fear is the mind killer. It is the little death.”
a)
One Flew over the Cuckoo’s Nest
b)
The Grapes of Wrath
c)
Tom Sawyer and Huckleberry Finn
d)
Dune
e)
The Firm
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