AP Statistics 2021 Free-Response Questions

2021

AP? Statistics

Free-Response Questions

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I.

Descriptive Statistics

x

=

1 n

xi

=

xi n

y^= a + bx

Formulas for AP Statistics

s=x

n

1 -

1

(

xi

-

x )=2

( xi - x )2

n -1

y= a + bx

= r

n

1 -

1

xi - sx

x

yi - sy

y

b

=

r

sy sx

II. Probability and Distributions

P ( A B)= P ( A) + P ( B) - P ( A B)

P(A |B)

=

P(A B) P(B)

Probability Distribution

Discrete random variable, X

If has a binomial distribution

with parameters n and p, then:

P ( X=

x=)

n x

px (1 -

p )n - x

where x = 0, 1, 2, 3, , n

If has a geometric distribution with parameter p, then:

P ( X= x=) (1 - p)x-1 p

where x = 1, 2, 3,

Mean

?X = E ( X ) = xi P ( xi )

? X = np

? X

=

1 p

Standard Deviation

X = ( xi - ?X )2 P ( xi )

= X np (1 - p)

X =

1- p p

III. Sampling Distributions and Inferential Statistics

Standardized test statistic:

statistic - parameter standard error of the statistic

Confidence interval: statistic ? (critical value)(standard error of statistic)

Chi-square statistic: 2 =

(observed - expected)2

expected

III. Sampling Distributions and Inferential Statistics (continued) Sampling distributions for proportions:

Random Variable

For one population:

p ^

? p^ = p

Parameters of Sampling Distribution

p^ =

p (1 - p)

n

Standard Error* of Sample Statistic

s p^ =

p^ (1 - p^ )

n

For two populations:

p^1 - p^2

? p^1- p^=2 p1 - p2 = p^1- p^2

( ) ( ) p1 1 - p1 + p2 1 - p2= sp^1- p^2

n 1

n2

p^1 (1 - p^1 ) + p^2 (1 - p^2 )

n 1

n2

When p1 = p2 is assumed:

( ) s p^1- p^2

=

p^ c

1 - p^c

1 n 1

+

1 n2

where

p^ c

=

X1 + X2 n1 + n2

Sampling distributions for means:

Random Variable

For one population:

X For two populations: X1 - X2

Parameters of Sampling Distribution

? =?

X

=

X

n

? =

X1 - X 2

?1 - ?2

=

X1 - X 2

12

+

2 2

n1 n2

Standard Error* of Sample Statistic

sX =

s n

s

X 1

=

- X2

s12 + s22 n1 n2

Sampling distributions for simple linear regression:

Random Variable

Parameters of Sampling Distribution

For slope: b

?b =

b

=

x

n

,

where x =

( xi - x )2

n

Standard Error* of Sample Statistic

sb = sx

s , n -1

where s =

( yi - y^i )2

n - 2

and sx =

( xi - x )2

n -1

*Standard deviation is a measurement of variability from the theoretical population. Standard error is the estimate of the standard deviation. If the standard deviation of the statistic is assumed to be known, then the standard deviation should be used instead of the standard error.

AP? Statistics 2021 Free-Response Questions Begin your response to QUESTION 1 on this page.

STATISTICS SECTION II Total Time--1 hour and 30 minutes 6 Questions

Part A Questions 1-5 Spend about 1 hour and 5 minutes on this part of the exam. Directions: Show all your work. Indicate clearly the methods you use, because you will be scored on the correctness of your methods as well as on the accuracy and completeness of your results and explanations.

1. The length of stay in a hospital after receiving a particular treatment is of interest to the patient, the hospital, and insurance providers. Of particular interest are unusually short or long lengths of stay. A random sample of 50 patients who received the treatment was selected, and the length of stay, in number of days, was recorded for each patient. The results are summarized in the following table and are shown in the dotplot.

Length of stay (days) Number of patients

5

6

7

8 9 12 21

4 13 14 11 6

1

1

(a) Determine the five-number summary of the distribution of length of stay.

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AP? Statistics 2021 Free-Response Questions Continue your response to QUESTION 1 on this page.

(b) Consider two rules for identifying outliers, method A and method B. Let method A represent the 1.5 ? IQR rule, and let method B represent the 2 standard deviations rule. (i) Using method A, determine any data points that are potential outliers in the distribution of length of stay. Justify your answer.

(ii) The mean length of stay for the sample is 7.42 days with a standard deviation of 2.37 days. Using method B, determine any data points that are potential outliers in the distribution of length of stay. Justify your answer.

(c) Explain why method A might identify more data points as potential outliers than method B for a distribution that is strongly skewed to the right.

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