AP Stats Final Exam Review

Suppose there is a correlation of r = 0.9 between number of hours per day students study and GPAs. Which of the following is a reasonable conclusion?

• 90% of students who study receive high grades.

• 90% of students who receive high grades study a lot.

• 90% of the variation in GPAs can be explained by variation in number of study hours.

• 10% of the variation in GPAs cannot be explained by variation in number of study hours per day.

• 81% of the variation in GPAs can be explained by variation in number of study hours per day.

Answer explanation

If r = 0.9, then r² = 0.81, which is the coefficient of determination.

The coefficient of determination is interpreted as:

[r² as a %] of the variation in the [y-variable] can be explained by variation in the [x-variable].

A pet food manufacturer runs an experiment to determine whether three brands of dog food are equally preferred by dogs. In the experiment, 150 dogs are individually presented with three dishes of food, each containing a different brand, and their choices are noted. Tabulations show that 62 dogs go to brand A, 43 to brand B, and 45 to brand C. Is there sufficient evidence to say that dogs have preferences among the brands? Test at the 10% significance level.

• No, with χ² = 2.09, there is not sufficient evidence even at the 25% significance level.

• No, with χ² = 4.36, there is not sufficient evidence at the 10% level.

• No, with χ² = 19.0, there is not sufficient evidence even at the 0.1% level.

• Yes, with χ² = 4.36, there is sufficient evidence at the 10% level.

• Yes, with χ² = 19.0, there is sufficient evidence even at the 0.1% level.

Answer explanation

This would be a Chi-Squared GOF Test.

Brand A: Observed - 62, expected - 50

Brand B: Observed - 43, expected - 50

Brand C: Observed - 45, expected - 50

Put the observed counts in L1 and the expected counts in L2. Then run a Chi-Squared GOF Test.

Remember, df = # categories - 1 = 3 - 1 = 2

χ2 = 4.36

p = 0.113

since p-value > alpha (0.1), no, there is not sufficient evidence.

The American Medical Association (AMA) wishes to determine the percentage of obstetricians who are considering leaving the profession because of the rapidly increasing number of lawsuits against obstetricians. How large a sample should be taken to find the answer to within ±3% at the 95% confidence level?

• 6

• 33

• 534

• 752

• 1068

Answer explanation

Using the equation ME = z* √(p̂(1-p̂)/n)

ME = 0.03

z* = invNorm(0.025, 0, 1) = 1.96

p̂ = 0.5 (most conservative valuation)

0.03 = 1.96√(0.5(0.5)/n)

0.03 = 1.96(0.5)/√n

√n = 1.96(0.5)/0.03

n = [1.96(0.5)/0.03]²

n = 1067.11 -> 1068

Which of the following are true statements?

I. If there is sufficient evidence to reject a null hypothesis at the 10% level, then

there is sufficient evidence to reject it at the 5% level.

II. Whether to use a one- or two-sided test is typically decided after the data

are gathered.

III. If a hypothesis test is conducted at the 1% level, there is a 1% chance of

rejecting the null hypothesis

• I

• II

• III

• None are true

Answer explanation

I. If there is sufficient evidence to reject at a given alpha-level, then there is sufficient evidence to reject at every alpha-level greater than the given. (remember: reject if p>ɑ)

II. The hypotheses, and thus the type of test, is determined prior to gathering the data and running the test

III. The chance of rejecting a true null hypothesis is equal to the alpha-level

A manufacturer claims that a particular automobile model will get 50 miles per gallon on the highway. The researchers at a consumer-oriented magazine believe that this claim is high and plan a test with a simple random sample of 30 cars. Assuming the standard deviation between individual cars is 2.3 miles per gallon, what should the researchers conclude if the sample mean is 49 miles per gallon?

• There is not sufficient evidence to reject the manufacturer’s claim; 49 miles per gallon is too close to the claimed 50 miles per gallon.

• The manufacturer’s claim should not be rejected because the P-value of 0.012 is too small.

• The manufacturer’s claim should be rejected because the sample mean is less than the claimed mean.

• The P-value of 0.012 is sufficient evidence to reject the manufacturer’s claim.

• The P-value of 0.012 is sufficient evidence to prove that the manufacturer’s claim is false.

Answer explanation

This is a 1-Sample T-Test for Means

μ = 50

x̄ = 49

Sx = 2.3

n = 30

Ho: μ = 50

Ha: μ < 50

t = -2.38

p = 0.012

Since p < alpha (0.05), reject Ho.

A historian believes that the average height of soldiers in World War II was greater than that of soldiers in World War I. She examines a random sample of records of 100 men in each war and notes standard deviations of 2.5 and 2.3 inches in World War I and World War II, respectively. If the average height from the sample of World War II soldiers is 1 inch greater than from the sample of World War I soldiers, what conclusion is justified from a two-sample hypothesis test where H₀: μ₁ - μ₂ = 0 and Hₐ: μ₁ - μ₂ < 0?

• The observed difference in average height is significant.

• The observed difference in average height is not significant.

• A conclusion is not possible without knowing the mean height in each sample.

• A conclusion is not possible without knowing both the sample means and the two original population sizes.

• A two-sample hypothesis test should not be used in this example.

Answer explanation

This is a 2-sample t-test for difference of means.

Since p-value (0.0018) < alpha (0.05), reject Ho and observed difference is significant.

Which of the following statements about the correlation coefficient are true?

I. The correlation coefficient and the slope of the regression line may have opposite signs. II. A correlation of 1 indicates a perfect cause-and-effect relationship between the variables. III. Correlations of +.87 and –.87 indicate the same degree of clustering around the regression line.

• I only

• II only

• III only

• I and II

• I, II, and III

Answer explanation

I. The correlation coefficient, r, and the slope always have the same sign.

II. Regression cannot show causation!!!

III. This is true. An r = 0.87 and r = -0.87 show the same degree of clustering, or strength, just in different directions.

A soft drink dispenser can be adjusted to deliver any fixed number of ounces of soft drink. If the machine is operating with a standard deviation in delivery equal to 0.3 ounces, what should be the mean setting so that a 12-ounce cup will overflow less than 1% of the time? Assume a normal distribution for ounces delivered.

• 11.23 ounces

• 11.30 ounces

• 11.70 ounces

• 12.70 ounces

• 12.77 ounces

Answer explanation

We know:

σ = 0.3

to overflow x = 12

area above x = 0.01

first, find z:

z = invNorm(area: 0.99, μ = 0, σ = 1) = 2.3263

Use z-score formula to solve for μ.

The death rate from a particular form of cancer is 23% during the first year. When treated with an experimental drug, only 15 out of 84 patients die during the initial year. Is this strong evidence to claim that the new medication reduces the mortality rate?

• Yes, because the p-value is .0459.

• Yes, because the p-value is .1314.

• No, because the p-value is only .0459.

• No, because the p-value is above .10.

• An answer cannot be given without first knowing if a placebo was also used and what the results were.

Answer explanation

This is a 1-sample z-test for proportions.

Ho: p = 0.23

Ha: p < 0.23

In calculator:

p_o = 0.23

x = 15

n = 84

z = -1.12

p = 0.131

Since p-value (0.131) > alpha (0.05), there is not sufficient evidence and fail to reject Ho

Consider the following back-to-back stemplot.

Which of the following are true statements?

I. The distributions have the same mean.

II. The distributions have the same range.

III. The distributions have the same standard deviation.

• I only

• I and II

• I and III

• II and III

• I, II, and III

Answer explanation

Put each side of the stemplot into L1 and L2, respectively. Then do 1-Var Stats on each list separately.

Stats for left side:

x̄ = 54.86

s = 25.03

range = 76

Stats for right side:

x̄ = 34.86

s = 25.03

range = 76

Distributions have the same standard deviation and range. Only the mean varies.