150
Math 156–Sat: HW #5
Name:
1. For each of the following claims, determine the null and alternative hypotheses. Also, state whether
the test is two-tailed, left-tailed, or right-tailed. (8 points each)
(a) The Medco pharmaceutical company has just developed a new antibiotic for children.
Among the competing antibiotics, 2% of children who take the drug experience headaches as
a side effect. A researcher from the FDA wishes to test the claim that the percentage of
children taking the new antibiotic who experience headaches as side effect is more than the
competition.
(b) The Blue Book Value of a used 3-year-old Chevy Corvette is $37500. A consumer advocate
group wishes to test the claim that the mean price of a used 3-year-old Chevy Corvette is not
$37500.
2. According to the Centers for Disease Control and Prevention, 16% of children aged 6 to 11 years of
age are overweight. A school nurse believes that in her school district the percentage of 6 to 11 years
olds who are overweight is higher than this, and she tests the relevant hypotheses. Explain what a
Type I and a Type II error would be in this case. (8 points)
3. What is meant by the “level of significance” for a hypothesis test?
(6 points)
4. For each P – value given, state the conclusion that would be made when performing a hypothesis test
at the 0.05 level. (2 points each)
(a) 0.001
(b) 0.021
(c) 0.078
(d) 0.047
5. For each of the scenarios described below, state the P-Value.
(a)
(b)
(e) 0.148
(5 points each)
(c)
H 0 : µ = 24.3
H 0 : ! = 0.48
H 0 : µ = 107.5
H a : µ < 24.3
H a : ! " 0.48
H a : µ > 107.5
Test Stat : x = 22.8
! = 8.2
n = 36
Test Stat : p = 0.4975
n = 50
Test Stat : x = 109.2
s = 4.8
n = 64
6. The drug Prevnar is a vaccine meant to prevent meningitis (it also helps control ear infections). It is
typically given to infants. In clinical trials, the vaccine was administered to 710 randomly selected
infants between the ages of 12 and 15 months. Of the 710 infants, 121 experienced a loss of appetite.
Is there significant evidence to conclude that the proportion of infants who receive Prevnar and
experience loss of appetite is different from 0.135, the proportion of children who experience a loss of
appetite when taking comparable medications? Test at the 0.01 level. (10 points)
7. In September 1996, 33% of Americans believed in haunted houses. In a recent poll conducted, 370 of
1002 randomly selected Americans aged 18 or older said they believed in haunted houses. Is there
sufficient evidence to claim that the proportion of Americans who believe in haunted houses has
increased? Test the relevant hypotheses at the 0.05 level. (10 points)
8. According to the U.S. Federal Highway Administration, the mean number of miles driven annually by
Americans is 12200. A researcher in Montana believes that residents of Montana drive more than the
national average. She obtains a simple random sample of 35 drivers and finds the mean driving
distance for a year is 12895.7 with a standard deviation of 1158.4 miles. Test the relevant hypotheses
at the 0.05 level. (10 points)
9. A researcher claims that the average height of a woman aged 20 years or older is greater than the
1994 mean height of 63.7 inches on the basis of data obtained from the Centers for Disease Control
and Prevention’s Advanced Data Report, No. 347. The researcher obtained a simple random sample
of 45 women and found the sample mean height to be 63.9 inches. Assuming the standard deviation
for the height of all women is 3.4 inches, test the relevant hypotheses at the 0.05 level to see if the
researcher’s claim is correct. (10 points)
10. A nutritionist claims that children under the age of 10 years are consuming more than the U.S. Food
and Drug Administration’s recommended daily allowance of sodium, which is 2400mg. To test this
claim, she obtains a random sample of 75 children under the age of 10 and measures their daily
consumption of sodium. The mean amount of sodium consumed was determined to be 2993mg with
a standard deviation of 1489mg. Is there significant evidence to support the claim of the nutritionist?
Test the relevant hypotheses at the 0.01 level. (10 points)
11. For each of the following scenarios, determine if the sampling is dependent (so a paired test would be
used) or independent (so a 2-sample test would be used). (5 points each)
(a) A sociologist wants to determine if, within married couples, the man or woman tends to have
the larger income. She obtains a random sample of 50 married couples in which both spouses
work and determines each spouse’s annual income.
(b) An educator wants to determine whether a new curriculum significantly improves test scores
for third-grade students. She randomly divides 80 third-graders into two groups. Group 1 is
taught using the new curriculum while group 2 is taught using the traditional curriculum. At
the end of the school year, both groups are given the standardized test and the mean scores
are compared.
(c) A psychologist wants to know whether subjects respond faster to a go/no go stimulus or a
choice stimulus. With the go/no go stimulus, subjects must respond to a particular stimulus
by pressing a button and disregard other stimuli. In the choice stimulus, the subjects respond
differently depending on the stimulus. She randomly selects 20 subjects and each subject is
presented a series of go/no go stimuli and choice stimuli. The mean reaction time to each
stimulus is compared.
12. Do people walk faster in the airport when they are arriving than when they are departing? Researcher
collected the data below. Test the relevant hypotheses at the 0.05. (10 points)
Departure
Arrival
Mean Speed (feet per minute)
260
269
Std. Deviation (feet per minute)
53
34
Sample Size
35
35
13. Octane is a measure of how much the fuel can be compressed before it spontaneously ignites. Some
people believe that higher-octane fuels result in better gas mileage for their car. To test this claim, a
researcher randomly selects 11 individuals (and their cars) to participate in the study. Each
participant received 10 gallons of 87-octane gas and 10 gallons of 92-octane gas and drove his car on
a closed course until the car ran out of gas. The number of miles driven for each type of gas was
recorded. The results are in the table below. Test at the 0.01 level whether higher-octane gives better
gas mileage. You may assume that gas mileages are normally distributed. (10 points)
Driver
Miles on
87-octane
Miles on
92-octane
1
2
3
4
5
6
7
8
9
10
11
234
257
243
215
114
287
315
229
192
204
547
237
238
229
224
119
297
351
241
186
209
562
14. In a study reported in the paper “Ginkgo for Memory Enhancement” (Journal of the American
Medical Association [2002]), elderly adults were assigned at random to either a treatment group or a
control group. Those assigned to the treatment group took 40mg of ginkgo 3 times a day for 6 weeks.
Those assigned to the control group took a placebo pill 3 times a day for 6 weeks. At the end of the 6
weeks, the Wechsler Memory Scale (a test of short-term memory) was administered. Higher scores
indicate better memory function. Summary data is below. Test at the 0.05 level whether taking
ginkgo seems to improve short term memory. (10 points)
Ginkgo
Placebo
n
104
115
x
5.65
5.5
s
0.6
0.6
Math 156–Sat: HW #5 Solutions
1. (a) H0: π = 0.02
Ha: π > 0.02
This is a right-tailed (or upper-tailed) test.
(b) H0: µ = 37500
Ha: µ ≠ 37500
This is a two-tailed test.
2. A Type I error would be for the nurse to conclude that in her district more than 16% of children aged
6 to 11 years of age are overweight when in fact this is not true. A type II error would be for the
nurse to conclude that she cannot say more than 16% of children aged 6 to 11 years of age in her
district are overweight when in fact when in fact it is true.
3. The level of significance is the probability of making a Type I error.
4. (a)
(b)
(c)
(d)
(e)
reject H0
reject H0
fail to reject H0
reject H0
fail to reject H0
5. (a) z =
(b) z =
(c) t =
x " µ0
!
n
=
p " !0
! 0 (1"! 0 )
n
x ! µ0
s
n
6. 1-PropZTest
H0:
=
22.8 " 24.3
8.2
36
=
# "1.0976 , so P-Value = normalcdf (!", ! 1.0976 ) = 0.1362
0.4975 " 0.48
0.48(0.52 )
50
109.2 ! 107.5
4.8
64
# 0.2477 , so P-Value = 2normalcdf (0.2477, ! ) " 0.8044
" 2.8333 with 63df, so P-Value = tcdf (2.8333, !, 63) " 0.0031
π = 0.135
Ha:
π ≠ 0.135
Test Stat:
p=
121
710
! 0.1704, z = 2.762
P-Value = 0.0057
Conclude: Reject H0 – The proportion of children who experience loss of appetite when taking
Prevnar does seem to be different than 0.135.
Validity:
nπ = 121 and n(1-π) = 589 (both are more than 10)
7. 1-PropZTest
H0:
Ha:
π = 0.33
π > 0.33
370
p = 1002
! 0.369, z = 2.643
Test Stat:
P-Value = 0.0041
Conclude: Reject H0 – The proportion of Americans who believe in haunted houses does seem
to have increased from 0.33.
Validity:
nπ = 370 and n(1-π) = 632 (both are more than 10)
8. TTest
H0:
µ = 12200
Ha:
µ > 12200
Test Stat:
x = 12895.7, t = 3.5530
P-Value = 0.00057
Conclude: Reject H0 – Drivers in Montana do seem to drive more than the national average of
12200 miles per year.
Validity:
n = 35, which is more than 30, so Central Limit Theorem applies
9. ZTest
H0:
µ = 63.7
Ha:
µ > 63.7
Test Stat:
x = 63.9, z = 0.3946
P-Value = 0.3466
Conclude: Fail to reject H0 – evidence does not support that the average height of women is
more than the 1994 average.
Validity:
n = 45, which is more than 30, so Central Limit Theorem applies
10. TTest
H0:
µ = 2400
Ha:
µ > 2400
Test Stat:
x = 2993, t = 3.44898
P-Value = 0.000466
Conclude: Reject H0 – Children under the age of 10 do seem to be consuming more sodium than
the U.S. FDA recommends daily.
Validity:
n = 75, which is more than 30, so Central Limit Theorem applies
11. (a) Dependent
(b) Independent
(c) Dependent
12. 2-SampTTest
H0:
µD = µA
Ha:
µD < µA
Test Stat:
x D " x A = 260 " 269 = "9, t = "0.8456
P-Value = 0.2006
Conclude: Fail to reject H0 – evidence does not support that people walk faster when arriving
!
than when departing.
Validity:
Both sample sized are > 30 (35 and 35), so Central Limit Theorem applies
13. Paired TTest
Let µd represent the average of (miles on 87-octane) – (miles on 92-octane)
H0:
µd = 0
Ha:
µd < 0
Test Stat:
xd = !5.0909, t = !1.1350
P-Value = 0.1414
Conclude: Fail to reject H0 – evidence does not support that one gets better gas mileage when
using 92-octane gas than when using 87-octane gas.
Validity:
given normally distributed populations
14. 2-SampTTest
H0:
µG = µP
Ha:
µG > µP
Test Stat:
xG ! x P = 5.65 ! 5.5 = 0.15, t = 1.8475
P-Value = 0.0330
Conclude: Reject H0 – Taking ginkgo does seem to improve short term memory.
Validity:
Both sample sized are > 30 (104 and 115), so Central Limit Theorem applies