Purpose:
This activity is intended to illustrate properties of hypothesis testing and to describe how to perform hypothesis tests on a proportion.
Statistical Guide: (Aliaga and Gunderson 1999)
We want to test hypotheses about the population proportion p. The null hypothesis is H_{o}:p=p_{o}, where p_{o} is the hypothesized value for p. The data are assumed to be a random sample of size n from the population, where n is large (n must be large enough so that np_{o}³5 and n(1p_{o})³5). From the sample data, we calculate the sample proportion
_{}.
We base our decision about p on the standardized sample proportion
_{}.
This zscore is the test statistic and its distribution under H_{o} is approximately N(0,1).
We calculate the pvalue (observed level of significance) for the test, which depends on how the alternative hypothesis is expressed:
(1) If H_{a}:p>p_{o}, then the pvalue is the area to the right of the observed test statistic under the H_{o} model.
(2) If H_{a}:p<p_{o}, then the pvalue is the area to the left of the observed test statistic under the H_{o} model.
(3) If H_{a}:p¹p_{o}, then the pvalue is the sum of the area to the left of negative the absolute value of the observed test statistic and the area to the right of the absolute value of the observed test statistic under the H_{o} model.
The pvalue is the probability, computed under the assumption that H_{o} is true, of obtaining a test statistic value at least as favorable to H_{a} as the value that actually resulted from the data. H_{o} is rejected if the pvalue is small enough.
Rejecting the null hypothesis when in fact it is true is called a Type I error. The significance level, a, is the chance of committing a Type I error. H_{o} is rejected if the pvalue £ a.
Failing to reject the null hypothesis when in fact it is not true is called a Type II error. The chance of committing a Type II error is b. The chance of rejecting the null hypothesis when in fact it is false is called the Power. The Power of a test is 1b.
Instructions:
Each person should have 10 plain KISSES^{® }chocolates, a 16ounce plastic cup, and two sticky notes.
Examine one of the KISSES^{® }chocolates. There are two possible outcomes when a KISSES^{® }chocolate is tossed  landing completely on the base or not landing completely on the base.
Estimate p, the proportion of the time that a KISSES^{® }chocolate will land completely on its base when
tossed: (circle one)
p is approximately 50% p < 50%
The investigation is as follows:
Put 10 KISSES^{® }chocolates into the cup; and:
· Gently shake the cup twice to help mix up the candies.
· Tip the cup so the bottom of the rim is approximately 12 inches from the table and spill the candies.
· Count the number of candies that land completely on their base.
· Return the candies to the cup and repeat until you have spilled the candies 5 times.
Record your results on the Data Table.
Data Table:
Toss Number 
Number of Candies Landing Completely on Base 
1 

2 

3 

4 

5 

Total 

Questions:
Throughout answering the following questions, assume that the true value of p = .35. This value is based on previous experience with tossing plain KISSES^{® }chocolates.
1. Test H_{o}:p=.5 versus H_{a}:p<.50.
Use pvalues (observed significance levels) to perform the tests. Take calculations to two significant digits. Note: H_{o} is false. A correct decision would be to reject H_{o}. An incorrect decision would be to fail to reject H_{o}. (This would be a Type II error.)
(a) Use all of your data for the tosses (n = 50) and a 5% level of significance (a = .05).
calculated test statistic =
pvalue =
(Write your pvalue on a sticky note and place it on the stemandleaf plot labeled Question 1.)
decision =
P(Type II error) = b. For the class b =
Power =1b. For the class 1b =
(b) Use all of your data for the tosses (n = 50) and a 20% level of significance (a = .20).
calculated test statistic =
pvalue =
decision =
P(Type II error) = b. For the class b =
Power = 1b. For the class 1b =
(c) Explain, using complete sentences, how to interpret a Type II error rate in terms of repeatedly performing the procedure of selecting a sample and using the sample data to test a hypothesis about a population parameter.
(d) Explain, using complete sentences, how the Type I error rate (a) is related to the Type II error rate (b). In addition, give an intuitive explanation as to why this relationship holds.
2. Test H_{o}:p=.35 versus H_{a}:p¹.35.
Use pvalues (observed significance levels) to perform the tests. Take calculations to two significant digits. Note: H_{o} is true. A correct decision would be to fail to reject H_{o}. An incorrect decision would be to reject H_{o}. (This would be a Type I error.)
(a) Use all of your data for the tosses (n = 50) and a 5% level of significance (a = .05).
calculated test statistic =
pvalue =
(Write your pvalue on a sticky note and place it on the stemandleaf plot labeled Question 2.)
decision =
expected number of rejections of H_{o} for the class =
number of rejections of H_{o} for the class =
(b) Use all of your data for the tosses (n = 50) and a 20% level of significance (a = .20).
calculated test statistic =
pvalue =
decision =
expected number of rejections of H_{o} for the class =
number of rejections of H_{o} for the class =
(c) Explain, using complete sentences, how to interpret a Type I error rate in terms of repeatedly performing the procedure of selecting a sample and using the sample data to test a hypothesis about a population parameter.
Stemandleaf plot for class pvalues:
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1 
Answers to Activity Questions and Assessment Questions:
Activity Questions.
1.
(c) If we repeatedly perform the procedure of selecting a sample and using the sample data to test a hypothesis about a population parameter, the Type II error rate is the percentage of the samples that would lead us to fail to reject a false null hypothesis.
(d) The Type I error rate is inversely related to the Type II error rate. If, for instance, we decrease the Type I error rate, we are making it more difficult to reject the null hypothesis, which in turn will increase the chances of failing to reject a false null hypothesis; therefore increasing the Type II error rate.
2.
(c) If we repeatedly perform the procedure of selecting a sample and using the sample data to test a hypothesis about a population parameter, the Type I error rate is the percentage of the samples that would lead us to reject a true null hypothesis.
Assessment Questions.
1.
(a) A Type I error in the context of this problem would be if the parachutist concludes that the parachute will not open when in fact it will open.
(b) A Type II error in the context of this problem would be if the parachutist does not conclude that the parachute will fail to open when in fact it will fail to open.
(c) A Type II error is far worse in this situation. A Type I error would lead the parachutist either not to jump or to get a new parachute, when the parachute she has is actually fine. The only loss would be time and or minimal financial loss. A Type II error would lead to the parachutist jumping and the parachute failing to open. This would mean serious injury or death.
(d) The backup guards against Type II error. If the parachutist jumps and the rip cord fails to open the parachute, then she can employ the backup and still land safely.
(e) If you were concerned about a Type I error you would pull the rip cord again. You wouldn't want to conclude the rip cord will fail when it will actually work. If you were concerned about a Type II error you would pull the backup. You wouldn't want to fail to use the backup if the rip cord is truly malfunctioning. I'd pull that backup. You've already tried the rip cord and it failed to deploy the parachute. I'm not about to give it another chance.
2.
(a) Type I error and Type II error are inversely related. Thus, if you increase the Power of a test (thereby decreasing the chance of a Type II error) this will increase the chance of a Type I error.
(b) If you set the Type I error rate at .0001 then you are increasing the chance of making a Type II error. That is, it will be very hard to reject a false null hypothesis.
3.
Each student has generated a value for _{}. A histogram of these values is an approximation to the distribution of the sample proportion _{}. The mean of these _{} values is an estimate of the mean _{}of the sampling distribution. The standard deviation of these _{} values is an estimate of the standard deviation _{}of the sampling distribution.
4.
(a) _{}
pvalue = _{} = .0793 Fail to Reject H_{o}, since .0793 > .05.
(b) _{}
pvalue = _{} = .0228 Reject H_{o}, since .0228 < .05.
(c) When you use a larger sample size, you decrease the standard error. This makes it easier to reject the null hypothesis.
(d) When the sample size increases, the standard error decreases, which results in the absolute value of the test statistic increasing.
(e) When the sample size increases, the absolute value of the test statistic increases, which results in the pvalue decreasing.
(f) As the sample size increases, the probability of a Type II error decreases, and the Power of the test will increase.