Question

question 3
in the united states, males between the ages of 40 and 49 eat on average 103.1 g of fat every day with a standard deviation of 4.32 g. assume that the amount of fat a person eats is normally distributed.
round the probabilities to four decimal places.
it is possible with rounding for a probability to be 0.0000.
a) state the random variable.
b) find the probability that a randomly selected male in the us between the ages of 40 and 49 has a fat consumption of 94.94 g or grams or more.
c) find the probability that a randomly selected male in the us between the ages of 40 and 49 has a fat consumption of 98.14 g or grams or less.
d) find the probability that a randomly selected male in the us between the ages of 40 and 49 has a fat consumption between 94.94 and 98.14 g or grams.
e) find the probability that randomly selected male in the us between the ages of 40 and 49 has a fat consumption that is at least 118.22 g or grams.
f) is a fat consumption of 118.22 g or grams unusually high for a randomly selected male in the us between the ages of 40 and 49?
why or why not?
g) what fat consumption do 51% of all males in the us between the ages of 40 and 49 have less than?
round your answer to two decimal places in the first box.
put the correct units in the second box.

question 4
a dishwasher has a mean lifetime of 12 years with an estimated standard deviation of 1.25 years. assume the lifetime of a dishwasher is normally distributed.
round the probabilities to four decimal places.
it is possible with rounding for a probability to be 0.0000.
a) state the random variable.
b) find the probability that a randomly selected dishwasher has a lifetime of 10.85 years or more.
c) find the probability that a randomly selected dishwasher has a lifetime of 11.05 years or less.
d) find the probability that a randomly selected dishwasher has a lifetime between 10.85 and 11.05 years.
e) find the probability that randomly selected dishwasher has a lifetime that is at most 8.875 years.
f) is a lifetime of 8.875 years unusually low for a randomly selected dishwasher?
why or why not?
g) what lifetime do 54% of all dishwashers have less than?
round your answer to two decimal places in the first box.
put the correct units in the second box.

Explanation:

Identify parameters for Question 3

Using the Normal Distribution knowledge point

$$ \mu = 103.1\,\text{g},\quad \sigma = 4.32\,\text{g} $$

Define random variable and calculate probabilities for Question 3

Using the Z-Score Calculation and Standard Normal Table knowledge points

• a) The random variable \(X\) is the fat consumption of a randomly selected male in the US between 40 and 49.
• b) For \(X \ge 94.94\):

$$ z = \frac{94.94 - 103.1}{4.32} = -1.89 \implies P(Z \ge -1.89) = 1 - 0.0294 = 0.9706 $$

• c) For \(X \le 98.14\):

$$ z = \frac{98.14 - 103.1}{4.32} = -1.15 \implies P(Z \le -1.15) = 0.1251 $$

• d) For \(94.94 \le X \le 98.14\):

$$ P(-1.89 \le Z \le -1.15) = 0.1251 - 0.0294 = 0.0957 $$

• e) For \(X \ge 118.22\):

$$ z = \frac{118.22 - 103.1}{4.32} = 3.50 \implies P(Z \ge 3.50) = 1 - 0.9998 = 0.0002 $$

• f) Yes, because the probability of a male consuming at least \(118.22\,\text{g}\) is very low (\(0.0002 \le 0.05\)).
• g) For the 51st percentile (\(P = 0.51\)):

$$ z \approx 0.03 \implies X = 103.1 + 0.03 \times 4.32 = 103.23\,\text{g} $$

Identify parameters for Question 4

Using the Normal Distribution knowledge point

$$ \mu = 12\,\text{years},\quad \sigma = 1.25\,\text{years} $$

Define random variable and calculate probabilities for Question 4

Using the Z-Score Calculation and Standard Normal Table knowledge points

• a) The random variable \(X\) is the lifetime of a randomly selected dishwasher.
• b) For \(X \ge 10.85\):

$$ z = \frac{10.85 - 12}{1.25} = -0.92 \implies P(Z \ge -0.92) = 1 - 0.1788 = 0.8212 $$

• c) For \(X \le 11.05\):

$$ z = \frac{11.05 - 12}{1.25} = -0.76 \implies P(Z \le -0.76) = 0.2236 $$

• d) For \(10.85 \le X \le 11.05\):

$$ P(-0.92 \le Z \le -0.76) = 0.2236 - 0.1788 = 0.0448 $$

• e) For \(X \le 8.875\):

$$ z = \frac{8.875 - 12}{1.25} = -2.50 \implies P(Z \le -2.50) = 0.0062 $$

• f) Yes, because the probability of a dishwasher lasting at most \(8.875\,\text{years}\) is very low (\(0.0062 \le 0.05\)).
• g) For the 54th percentile (\(P = 0.54\)):

$$ z \approx 0.10 \implies X = 12 + 0.10 \times 1.25 = 12.13\,\text{years} $$

Answer:

Question 3

a) State the random variable:
The fat consumption of a randomly selected male in the US between the ages of 40 and 49.

b) Find the probability of 94.94 g or more:
0.9706

c) Find the probability of 98.14 g or less:
0.1251

d) Find the probability between 94.94 and 98.14 g:
0.0957

e) Find the probability of at least 118.22 g:
0.0002

f) Is a fat consumption of 118.22 g unusually high?
Yes, because the probability of a male consuming this amount of fat or more is less than or equal to 0.05.

g) What fat consumption do 51% of all males have less than?
First box: 103.23
Second box: grams

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Question 4

a) State the random variable:
The lifetime of a randomly selected dishwasher.

b) Find the probability of 10.85 years or more:
0.8212

c) Find the probability of 11.05 years or less:
0.2236

d) Find the probability between 10.85 and 11.05 years:
0.0448

e) Find the probability of at most 8.875 years:
0.0062

f) Is a lifetime of 8.875 years unusually low?
Yes, because the probability of a dishwasher lasting this long or less is less than or equal to 0.05.

g) What lifetime do 54% of all dishwashers have less than?
First box: 12.13
Second box: years