MATH&146

Exam 2 Review

Lessons 12 – 23

1

Example 1

Explain to someone who has not taken a statistics

course what statistical significance means.

One answer:

A result is considered statistically significant if that

result's difference from the null hypothesis could

not plausibly have come from chance alone.

2

Example 2

A researcher conducts a study with a small sample

size (i.e., n is small). Is the researcher at a greater

risk of making a Type 1 Error or a Type 2 Error, or

are they equally likely. How do you know?

3

Test conclusion

do not reject H0 reject H0 in favor of HA

Truth

H0 trueCorrect Outcome

(True Negative)

Type 1 Error

(False Positive)

HA trueType 2 Error

(False Negative)

Correct Outcome

(True Positive)

Example 3

Food inspectors sample food products to see if they

are safe. This can be thought of as a hypothesis test

where

H0: The food is safe.

HA: The food is not safe.

a) What would be Type 1 Error and a Type 2 Error for

these hypotheses?

b) From the consumers' standpoint, would it be worse

for the inspector to make a Type 1 Error or a Type

2 Error? What about the producers' standpoint?4

Example 4

According to Harper's magazine, the time spent by

kids in front of the television set per year can be

modelled by a normal distribution with a mean equal to

1500 hours and a standard deviation equal to 250

hours.

a) What percent of kids watch television for less than

1200 hours per year?

b) What percent of kids watch for more than 1400

hours per year?

c) How much television do the top 25% of kids watch

each year?

5

Example 5

Suppose the true population proportion is p = 0.95.

Compute the expected number of successes and

failures for each sample size below. Are the conditions

for normality satisfied?

6

n = 20

n = 500

n = 100

Example 6

Can we approximate poker winnings below by a

normal distribution? Assume the winnings are

independent of each other.

7

Example 7

A card company claims that 80% of all American

college students send a card to their mother on

Mother's Day. Suppose you plan to gather your own

data to test this claim. You select a simple random

sample of 80 students to determine the proportion that

sent a card to their mother. Your sample indicates that

56 of the students sampled sent a card.

8

Example 7 continued

a) Write the hypotheses for testing the claim that 80%

of college students send their mothers a Mother's

Day card.

b) Calculate the proportion of students in this sample

that sent their mother a Mother's Day card.

c) Describe how to perform a simulation and, once

you had the results, how to estimate the p-value.

9

Example 7 continued

d) Below is a dot plot showing the distribution of

in 100 simulations under the null-hypothesis.

Estimate the p-value using the plot and determine

the conclusion of the hypothesis test.

10

simp̂

Example 8

A researcher wanted to assess whether his dog

Sophie tends to chase one of her toys more often than

the other. The researcher rolled both a green ball and

a red ball at the same time and observed which ball

Sophie chased. Repeating this process 100 times, he

found that Sophie chased the green ball 57 times and

the red ball 43 times.

a) Construct hypotheses regarding whether or not

Sophie preferred one ball over the other.

b) What proportion of the time in the sample did

Sophie choose the green ball?

11

Example 8 continued

c) Have the conditions to use the normal distribution

for inference been met?

d) Assuming the conditions have been met, compute

the standard error and p-value.

e) Write a suitable conclusion in the context of this

problem.

f) Create a 95% confidence interval to estimate the

proportion of times Sophie chooses the green ball

over the red ball. Are the results from the interval

consistent with your conclusion in part (e)?

12

Example 9

Two types of medication for hives are being tested to

determine if there is a difference in the percentage of

adult patient reactions. Twenty out of a random

sample of 200 adults given medication A still had hives

30 minutes after taking the medication. Twelve out of

another random sample of 200 adults given

medication B still had hives 30 minutes after taking the

medication. Test at a 1% level of significance.

13

Medication A B Total

Hives 20 12 32

No hives 180 188 368

Total 200 200 400