Additional Exercises on Hypothesis Testing
2200
Part A: Determine whether each example would best be tested using a hypothesis test on a) a single mean
from a single sample, b) a single mean from paired data (dependent samples), c) two means from
independent samples, d) single proportion from a single sample, or e) two proportions from two
independent samples. Then set up an appropriate null and alternative hypothesis for each example. (In
part B below, you will provided with data and will be asked to finish the hypothesis tests.)
1. The GRE is a test used for admission to graduate schools across the nation. A researcher in interested
in determining if having a person leading review sessions significantly improves scores over a self study
program. Students were randomly assigned to one of two groups. The same study material was used in
both groups, but in one group a qualified person reviewed with the students while in the other group, the
students studied on their own.
2. A biologist is interested in determining if, on average, soil inhabited by fire-ants has a different level of
iron than soil not inhabited by fire-ants. Samples of soil are taken from eight randomly selected, well
established, fire-ant mounds and also from seven randomly selected areas not to close to the fire-ant
mounds. For each sample the amount of iron is determined.
3. In the city of Slumlord, the local newspaper reports the mean rent of a two bedroom apartment is $645.
Since you recently were looking for an apartment, you feel this value is underestimating the true mean
rent of a two bedroom apartment in Slumlord.
4. It is believed that one of the factors that can increase women’s chance of having a multiple baby birth
is mother’s age. Each woman in the study was put into one of two categories 1) aged 34 or younger, or 2)
age 35 or older. Each delivery was then recorded as a single birth or multiple births. Is there evidence
suggesting that older mothers are more likely to give birth to multiples than younger mothers?
5. A highway patrolman wonders if the mean speed of passing motorists on a given stretch of highway is
different for weekday mornings vs. weekend mornings. To test, he checks the speeds of passing cars
during the weekend mornings and also during weekday mornings.
6. It is belied that at least 20% of graduating students at a university took one or more classes online. To
test this claim a sample of graduating students at the university are asked if they took any online classes.
Part B: Use the given data to finish the hypothesis test stated in Part A above. Use the specified level of
significance
1. Descriptive statistics from the GRE test scores are given for the two groups. Use a level of
significance of 0.025.
mean standard deviation
sample size
Self study scores:
500
85
35
With reviewer:
520
90
30
2. For each sample the amount of iron was determined. Use a level of significance of 0.10.
Iron content from fire-ant mounds: 4.8, 8.7, 13.9, 5.4, 13.8, 10.9, 16.3, 11.4
Iron content from surrounding soil: 13.3, 22.6, 10.4, 18.6, 14.3, 10.4, 26
3. From a random sample of 35 two bedroom apartments in Slumlord, the mean rent is calculated to be
$642 with a standard deviation of $47. Conduct the hypothesis test using a level of significance of 0.025.
4. The study included 752 women who gave birth during the past year. Of the 752 women, 531 were
aged 34 or less and among them 15 had multiple births; of the remaining 221 women who were 35 or
older, 12 had multiple births. Use a level of significance of 0.05.
5. Descriptive statistics on the speeds of passing motorists on weekends and on weekdays were calculated.
Use a level of significance of 0.05. (data is in miles per hour)
Mean
standard deviation
sample size
Weekday
32.1
2.8
45
Weekend
29.4
3.1
42
6. From a random sample of 120 graduating students, it is determined that 19 took an online class. Use a
level of significance of 0.05
Answers to Additional Exercises on Hypothesis Testing
Part A:
1. Two means from two independent samples: H0: µ reviewer = µ self study HA: µ reviewer > µ self study where
µ reviewer = represents the population mean GRE score when a reviewer is available and µ self study
represents the population mean GRE score when students study by themselves.
2. Two means from independent samples: H0: µ Fire-ant = µ surrounding soil HA: µ Fire-ant ≠ µ surrounding soil
where µ Fire-ant represents the mean amount of iron in the soil of fire ant hills, and µ surrounding soil
represents the mean amount of iron in the soil surrounding fire ant hills.
3. Single mean from a single sample: H0: µ = 645 HA: µ > 645 where µ represents the mean rent of a
two bedroom apartment
4. Two proportions from two independent samples: : H0: P older = P younger HA: P older > P younger where
P older represents population proportion of births by women (35 or older) that result in multiple babies,
and P younger represents population proportion of births by younger women (less than 35 years old) that
result in multiple babies.
5. Two means from independent samples: H0: µ weekday = µ weekend = HA: µ weekday ≠ µ weekend where µ
weekday represents the population mean speed of motorists on a weekday in the morning, and µ weekend
represents the population mean speed of motorists on the weekend in the morning.
6. One proportion from a single sample: H0: P > 0.20 HA: P < 0.20
proportion of graduates who took an online class.
where P represents population
Part B:
1. Statistics: t = 0.916, d.f. = 60.25, p-value = .1816;
Decision: with a level of significance of 0.025, fail to reject H0;
Statement: There is insufficient evidence to statistically claim the mean GRE score improves when a
person leads the review sessions versus letting students study on their own.
2. Statistics: x fire ant = 10.65
s fire ant = 4.12 n fire ant = 8
x surrounding = 16.51
s surrounding = 6.074 n surrounding = 7
t = -2.15, d.f. = 10.4, p-value = .055;
Decision: since the p-value of 0.055 is less than the level of significance of 0.10, reject H0;
Statement: From the sample data we can claim that, on average, the iron content in Fire-ant mounds
differs significantly from the iron content in surrounding soils. (It appears Fire-ants mounds have a lower
level of iron.)
3. Statistics: t = -0.38, p-value = 0.65
Decision: Fail to reject Ho.
Statement: It appears that is not enough information to support the claim that the mean rent of a two
bedroom apartment in Slumlord is greater than $645 per month.
4. p̂older = 0.054, p̂ younger = 0.028, z = 1.75, p-value 0.040, with a level of significance of 0.05 reject H0.
Statement: It appears there is sufficient evidence to support the fact that the proportion of multiple births
from older women (age 35 or older) is significantly higher than the proportion of multiple births from
younger women (age 34 or younger).
5. Statistics: t = 4.25, d.f. = 82.6, p-value 0.00005;
Decision: with a p-value = 0.00005, which is less than 0.05, reject H0.
Statement: There is statistically significant evidence to support the theory that the mean speed of cars on
that stretch of highway differ from weekday mornings to weekend morning. (The mean speed appears to
be significantly higher on the weekday mornings).
6. =0.158, z = -1.14, p-value 0.127, with alpha = 0.05, fail to reject H0.
Statement: There is no statistically significant evidence to claim that fewer than 20% of graduates at this
university took an online class.