Exam
Name___________________________________

SHORT ANSWER.  Write the word or phrase that best completes each statement or answers the question.
1) A biology professor reports that historically grades in her introductory biology course have been distributed as follows 15% A's, 30% B's, 40% C's, 10% D's, and 5% F's. Grades in her most recent course were distributed as follows

a. Test an appropriate hypothesis to decide if the professor's most recent grade distribution matches the historical distribution. Give statistical evidence to support your conclusion.
b. Which grade impacted your decision the most? Explain what this means in the context of the problem.	1)  _____________

2) In a local school, vending machines offer a range of drinks from juices to sports drinks.  The purchasing agent thinks each type of drink is equally favored among the students buying drinks from the machines. The recent purchasing choices from the vending machines are shown in the table.

a. Test an appropriate hypothesis to decide if the purchasing agent is correct. Give statistical evidence to support your conclusion.
b. Which type of drink impacted your decision the most? Explain what this means in the context of the problem.	2)  _____________

3) When two competing teams are equally matched, the probability that each team wins any game is 0.5. The NBA championship goes to the team that wins four games in a best-of-seven series. If the teams were equally matched, the probability that the final series ends with one of the teams sweeping four straight games would be 2 = 0.125. Further probability calculations indicate that 25% of these series should last five games, 31.25% should last six games, and the other 31.25% should last the full seven games. The table shows the number of games it took to decide each of the last 57 NBA champs. Do you think the teams are usually equally matched? Give statistical evidence to support your conclusion.

		3)  _____________

1) a. We want to know if the most recent grade distribution matches the historical grades distribution.
 The most recent grade distribution matches the historical grade distribution.
 The most recent grade distribution differs from the historical grade distribution.
Conditions
*Counted data We have the counts of the number of students who earned each grade.
*Randomization We have a convenience sample of students, but no reason to suspect bias.
*Expected cell frequency There are a total of 325 students. The smallest percentage of expected grades are F's, and we expect 325(0.05) = 16.25. Since the smallest expected count exceeds 5, all expected counts will exceed 5, so the condition is satisfied.

Under these conditions, the sampling distribution of the test statistic is  with 5 - 1 = 4 degrees of freedom, and we will perform a chi-square goodness-of-fit test.
		
 = 
=  +  +  +  +  = 62.538

The P-value is the area in the upper tail of the  model with 4 degrees of freedom above the computed  value.  P-value = P( > 62.538) = 8.5 × 
The P-value is very small, so we reject the null hypothesis. There is strong evidence that the most recent grade distribution differs from the historical grade distribution.

b. The  component for A's is the largest, 33.232. More students earned A's than we would have expected based on the historical grade distribution.
2) 

a. We want to know if the types of drinks are uniformly distributed (equally favored) among the students buying drinks from the machines.
 The types of drinks are uniformly distributed (equally favored) among the students buying drinks.
 The types are not uniformly distributed (equally favored) among the students buying drinks.
Conditions
* Counted data We have the counts from a sample of purchasing data.
* Randomization We don't want to make inferences about other high schools, so we don't need to check this condition.
* Expected cell frequency The null hypothesis expects 25% of the 680 drinks, or 170, should 	occur in each flavor. These expected values are all greater than 5, so the condition is satisfied.
Under these conditions, the sampling distribution of the test statistic is  with 4 - 1 = 3 degrees of freedom. We will perform a chi-square goodness of fit test.
 = =  +  +  +  = 8.012
The P-value is the area in the upper tail of the  model for 3 degrees of freedom above the computed  value.  P = P ( > 8.012) = 0.0458
The P-value of 0.0458 is low, so we reject the null hypothesis. These data show evidence that the drink flavors are not uniformly chosen (equally favored) by the students.
b. Kiwi Strawberry; its standardized residual is the largest  =  = 2.15, and its  component was 4.612.
More students preferred the Kiwi Strawberry compared to what was expected by the purchasing agent. The purchasing agent should order more Kiwi Strawberry than other  flavors.
3) 

 The distribution of length of series is consistent with a 50-50 chance for each team to win each game.
 The distribution of lengths of series isn't consistent with a 50-50 chance for each team to win each game.
Conditions
*Counted data These are counts of categorical data.
*Randomization We will assume that these 57 series are typical of past and future NBA championships.
*Expected cell frequency The expected values are all greater than five.
Under these conditions, we can perform a  goodness-of-fit test with 3 degrees of freedom.
 =  =  +  +  +  = 1.54
P = 0.67. The high P-value means we do not reject the null hypothesis; there is no evidence that the NBA championship series are inconsistent with the conjecture that the teams are evenly matched.