AP Statistics 2.11: Normal Interval Probabilities

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.

Model Solution