AP Statistics 2.11 The Normal Distribution Questions

Part (a):
No, it is not reasonable to believe that the distribution of 40-yard running times is approximately normal, because the minimum time is only 1.33 standard deviations below the mean $\left(z=\frac{4.4-4.6}{0.15} \approx-1.33\right)$. In a normal distribution, approximately 9.2 percent of the z-scores are below -1.33. However, there are no running times less than 4.4 seconds, which indicates that there are no running times with a z-score less than -1.33. Therefore, the distribution of 40 -yard running times is not approximately normal.

Official holistic scoring rubric

Parts (a), (b-i), and (b-ii) are each classified as essentially correct (E), partially correct (P), or incorrect (I). Minor arithmetic errors are not penalized.

Part (a): E requires all three components: use of a normal distribution with the correct parameter values, the correct boundary value, and the correct normal probability consistent with those choices. P requires exactly two components. Otherwise the part is I. A statistical-notation error lowers the part by one level, and acceptable correct probabilities range from 0.1020 through 0.1038.

Part (b-i): E requires a correct expected-value equation and correct solution, or another correct numerical procedure for the expected weight of one egg. P applies when the correct procedure is indicated but expected-value properties are applied incorrectly, or when the procedure is communicated poorly. Otherwise the part is I.

Part (b-ii): E requires combining variances and all three components: subtracting variances, correctly using 12 in the calculation, and reporting the corresponding correct standard deviation. P requires combining variances and exactly two components. Otherwise the part is I.

Overall score: 4 = all three parts E. 3 = two parts E and one part P, or one part E and two parts P. 2 = two parts E and one part I, one part E with one part P and one part I, or all three parts P. 1 = one part E and two parts I, or one or two parts P.

B

Random variable A, which represents the amount of shampoo in a randomly selected bottle, follows a normal distribution with mean 0.6 liter and standard deviation 0.04 liter. Then, the probability that a randomly selected bottle is underfilled is
$$P(A<0.5)=P\left(Z<\frac{0.5-0.6}{0.04}=-2.5\right) \approx 0.0062.$$

Scoring
Essentially correct (E) if the response includes the following three components:

Indicates the use of a normal (or approximately normal) distribution and identifies the correct parameter values (mean 0.6 and standard deviation 0.04)

Specifies the correct event (boundary value and direction), or an event consistent with values reported in component 1

Provides the correct probability of 0.0062 or probability consistent with components 1 and 2

Partially correct (P) if the response satisfies only two of the three components OR if the response fails to satisfy component 1 and 2, but shows the correct z-score formula, z-score value, and correct probability (e.g., $\frac{0.5-0.6}{0.04}=-2.5$, resulting in a probability of 0.0062).

Incorrect (I) if the response does not satisfy the criteria for E or P.

Additional Notes:
Component 1

- A response may satisfy component 1 by any of the following or a combination of the following:

Graphical: Displaying a graph of a normal density function with the appropriate scale on the horizontal axis showing the mean and standard deviation for the distribution of shampoo amount.

Calculator function syntax: Labeling correct values of the mean and standard deviation in a "normalcdf" statement, such as normalcdf (lower $=-\infty$, upper =0.5, mean =0.6, standard deviation =0.04 ).
Correct specification of the upper and lower bounds is not required to satisfy component 1.

Words: Using a statement such as "normal distribution with mean 0.6 and standard deviation 0.04."

Standard Notation: Using standard notation such as N(0.6,0.04) or $N\left(0.6,(0.04)^{2}\right)$.

Z-score: Displaying the correct mean and standard deviation in a z-score calculation that includes "z," such as $z=\frac{0.5-0.6}{0.04}$.

Component 2

- A response may satisfy component 2 by any of the following or a combination of the following:

Graphical: Displaying a graph of a normal density function with the region of interest ( A<0.5 or Z<-2.5 ) clearly identified. The shaded area does not need to be proportional, but the boundary should be on the proper side of the mean, and the shading should be in the proper direction.

Calculator function syntax: Identifying the lower and upper bounds of the region of interest in a "normalcdf" statement, such as:

- normalcdf (lower $=-\infty$, upper =0.5, mean =0.6, standard deviation =0.04 )

- normalcdf (lower $=-\infty$, upper $=-2. \bar{a},=0, \sigma=1)$

Correct specification of the mean and standard deviation is not required to satisfy component 2.

Words: Specifying the correct event in words with correct numerical values for the boundary value and correct direction, such as "the probability that the amount of shampoo is less than 0.5 liter" or P( amount of shampoo < 0.5).

Standard Notation: Using standard notation such as: P(A<0.5) or $P\left(z<\frac{0.5-0.6}{0.04}\right)$ or P(Z<-2.5).

General

- It is not necessary to define the random variable A because it is defined in the stem. It is not necessary to define the random variable Z because it is standard notation. Any other random variable must be defined correctly.

- An error in statistical notation, such as using s instead of $\sigma$ for the population standard deviation or using $\bar{x}$ instead of $\mu$ for the population mean, does not satisfy component 1.

- If the only error in the response to part (a) is the reversal of the numerator for the z-score (0.6-0.5), the response is scored P.

- An arithmetic or transcription error in a response can be ignored if correct work is shown.

Part (a):
The 25 th percentile of the standard normal distribution is -0.6745. Consequently the 25 th percentile of a normal distribution with mean 30 months and standard deviation 8 months is 30+8(-0.6745)=24.6 months.

B