ap stats unit 4 part 2 (6.1-6.8) test v1 name chase colma 1. the distribution of f, the number of cups of fluid that a randomly selected adult woman drinks per day, is shown in the table. f | 6 | 7 | 8 | 9 | 10 | 11 | 12 probability | 0.10 | 0.17 | 0.23 | 0.24 | 0.15 | 0.08 | 0.03 a. find the probability that a randomly selected adult woman drinks eleven cups of fluid per day. b. find and interpret the expected value of f. c. find the probability that a randomly selected adult woman drinks at least eight cups of fluid per day. d. the u.s. national academies of sciences, engineering, and medicine has determined that an adequate daily fluid intake for adult women is about 11.5 cups per day. find the probability that a randomly selected adult woman drinks at least the recommended amount of fluid per day given that they drink at least eight cups of fluid per day. show your work.

Question

Accepted Answer

### Part (a)
# Explanation:
## Step1: Recall total probability rule
The sum of all probabilities in a probability distribution is $ 1 $. So, to find $ P(F = 11) $, we subtract the sum of the other probabilities from $ 1 $.
Sum of known probabilities: $ 0.10 + 0.17 + 0.23 + 0.24 + 0.15 + 0.03 = 0.92 $
## Step2: Calculate $ P(F = 11) $
$ P(F = 11)=1 - 0.92 = 0.08 $
# Answer: $ 0.08 $

### Part (b)
# Explanation:
## Step1: Recall expected value formula
For a discrete random variable $ X $ with values $ x_i $ and probabilities $ p_i $, the expected value $ E(X)=\sum x_i p_i $
## Step2: Calculate each term
- $ 6\times0.10 = 0.6 $
- $ 7\times0.17 = 1.19 $
- $ 8\times0.23 = 1.84 $
- $ 9\times0.24 = 2.16 $
- $ 10\times0.15 = 1.5 $
- $ 11\times0.08 = 0.88 $
- $ 12\times0.03 = 0.36 $
## Step3: Sum the terms
$ E(F)=0.6 + 1.19 + 1.84 + 2.16 + 1.5 + 0.88 + 0.36 = 8.53 $
Interpretation: If we randomly select many adult women, the average number of cups of fluid they drink per day will be approximately $ 8.53 $ cups.
# Answer: The expected value is $ 8.53 $. It means the average number of cups of fluid a randomly selected adult woman drinks per day (over many trials) is $ 8.53 $ cups.

### Part (c)
# Explanation:
## Step1: Identify "at least 8"
"At least 8" means $ F\geq8 $, so we sum the probabilities for $ F = 8,9,10,11,12 $
## Step2: Calculate the sum
$ 0.23 + 0.24 + 0.15 + 0.08 + 0.03 = 0.73 $
# Answer: $ 0.73 $

### Part (d)
# Explanation:
## Step1: Recall conditional probability formula
$ P(A|B)=\frac{P(A\cap B)}{P(B)} $. Here, $ A $: drinks at least $ 11.5 $ cups (so $ F\geq12 $ since $ F $ is integer), $ B $: drinks at least $ 8 $ cups ($ P(B) = 0.73 $ from part (c))
## Step2: Find $ P(A\cap B) $
Since $ A $ is a subset of $ B $, $ P(A\cap B)=P(A)=P(F = 12)=0.03 $
## Step3: Calculate $ P(A|B) $
$ P(A|B)=\frac{0.03}{0.73}\approx0.0411 $ (or using the initial wrong approach correction: Wait, the recommended is $ 11.5 $, so $ F\geq12 $ (since $ F $ is number of cups, integer). So correct calculation: $ P(F\geq12|F\geq8)=\frac{P(F\geq12)}{P(F\geq8)}=\frac{0.03}{0.73}\approx0.041 $)
# Answer: Approximately $ 0.041 $ (or $ \frac{3}{73}\approx0.041 $)

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QUESTION IMAGE

Question

Explanation:

Part (a)

Step1: Recall total probability rule

Step2: Calculate \( P(F = 11) \)

Step1: Recall expected value formula

Step2: Calculate each term

Step3: Sum the terms

Step1: Identify "at least 8"

Step2: Calculate the sum

Answer:

Part (b)