Download 7.4 – Hypothesis Testing – Proportions – Notes KEY

Probability & Statistics
Testing Proportions
Mr. Coppock
NAME:___________________________________
The next hypothesis test that we will be looking at is the “1 sample z test for a proportion.”
Example: Brian and Matt are flipping a coin and betting on the results of heads or tails. Brian gets a
dollar from Matt whenever heads comes up and Matt gets a dollar from Brian whenever tails comes
up. At he end of 50 flips Brian is down $10 (in other words there are 20 heads and 30 tails in the 50
flips). Brian suspects that the coin is not balanced and that the proportion of heads is less than 0.5.
Do a hypothesis test (alpha = .05) to test Brian’s claim:
Step 1: State your hypothesis
p = the proportion of heads when the coin is flipped
Step 2: Assumptions
1) Data is an SRS from the population of interest.
2) The population is at least 10 times as large as the sample.
3) Where H o : p  po ; npo  5 , n(1  p0 )  5
Checking our assumptions:
1) Flipping a coin is like taking a random sample of all possible coin flips (each flip is independent
of the next)
2) The population of all coin flips is infinite, so it is more than 10 times as large as our sample size
of 50 flips.
3)
npo  50  0.5  25  5
n(1  p0 )  50(1  0.5)  25  5
Step 3: Calculate the test statistic and p value:
Z
pˆ  p o
p o (1  p o )
n
where: po  Hypothesized population proportion (from null hypothesis)
p̂ = Proportion found in our sample
In our case:
p o  0.5
pˆ  20 / 50  0.4
so:
Z
pˆ  p o
p o (1  p o )
n

0.4  0.5
0.5(1  0.5)
50

 0.1
0.25
50
 1.414
Find critical value
One-tailed test. Alpha is 0.05.
Critical value is -1.645.
Step 4: Conclusion
Since our test statistics is greater than the critical value, we do not have enough evidence to reject
the null hypothesis. This means we do not have enough evidence that the coin is favorable toward
tails.
Practice:
In a study of air-bag effectiveness, it was found that in 821 crashes of midsize cars equipped with
air bags, 46 of the crashes resulted in hospitalization of the drivers. Use a 0.01 significance level to
test the claim that the air-bag hospitalization rate is lower than the 7.8% rate for crashes of midsize
cars equipped with automatic safety belts.
2. In a consumer taste test, 100 regular Pepsi drinkers are given blind samples of Coke and Pepsi;
48 of these subjects preferred Coke. At the 0.05 level of significance, test the claim that Coke is
preferred by 50% of Pepsi drinkers who participate in such blind taste tests.
3. In 1990, 5.8% of job applicants who were tested for drugs failed the test. At the 0.01 level, test
the claims that the failure rate is now lower if a random sample of 1,520 current job applicants
results in 58 failures. Does the result suggest that fewer job applicants now use drugs?
4. Columbia Pictures claims that 58% of movies are R-rated. On a recent weekend, a survey of
movies at several area theaters found that 31 out of 53 movies were R-rated. At a 0.01 level of
significant, test the claim from Columbia Pictures.
HW: pp 392-393: #2, 3, 5, 7