Question

1/ Assume that the readings at freezing on a batch of thermometers are normally distributed with a mean of 0°C and a standard deviation of 1.00°C. A single thermometer is randomly selected and tested. Find P₃₁, the 31-percentile. This is the temperature reading separating the bottom 31% from the top 69%.
P₃₁ = °C
(Round answer to three decimal places)

2/ Engineers must consider the breadths of male heads when designing helmets. The company researchers have determined that the population of potential clientele have head breadths that are normally distributed with a mean of 6.9-in and a standard deviation of 1.2-in. Due to financial constraints, the helmets will be designed to fit all men except those with head breadths that are in the smallest 1.7% or largest 1.7%.
What is the minimum head breadth that will fit the clientele?
min =
What is the maximum head breadth that will fit the clientele?
min =
Enter your answer as a number accurate to 1 decimal place. Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

3/ The amounts of nicotine in a certain brand of cigarette are normally distributed with a mean of 0.958 g and a standard deviation of 0.322 g. Find the probability of randomly selecting a cigarette with 0.314 g of nicotine or less.
P(X < 0.314 g) =
Enter your answer as a number accurate to 4 decimal places. NOTE: Answers obtained using exact z-scores or z-scores rounded to 3 decimal places are accepted.

4/ In the country of United States of Heightlandia, the height measurements of ten-year-old children are approximately normally distributed with a mean of 56.5 inches, and standard deviation of 7 inches.
What is the probability that the height of a randomly chosen child is between 55.4 and 68.2 inches? Do not round until you get your your final answer, and then round to 3 decimal places.
Answer= (Round your answer to 3 decimal places.)

5/ A manufacturer knows that their items have a normally distributed lifespan, with a mean of 6.9 years, and standard deviation of 2.1 years.
If you randomly purchase one item, what is the probability it will last longer than 4 years?
Round answer to three decimal places

Answer 1

1) P(Z < z) = 0.31

Or, z = -0.496

Or, (X - )/ = -0.496

Or, (X - 0)/1 = -0.496

Or, X = -0.496

2) P(X < x) = 0.017

Or, P((X - )/ < (x - )/) = 0.017

Or, P(Z < (x - 6.9)/1.2) = 0.017

Or, (x - 6.9)/1.2 = -2.120

Or, x = -2.12 * 1.2 + 6.9

Or, x = 4.4

Min = 4.4

P(X > x) = 0.017

Or, P((X - )/ > (x - )/) = 0.017

Or, P(Z > (x - 6.9)/1.2) = 0.017

Or, P(Z < (x - 6.9)/1.2) = 0.983

Or, (x - 6.9)/1.2 = 2.120

Or, x = 2.12 * 1.2 + 6.9

Or, x = 9.4

Max = 9.4

3) P(X < 0.314)

= P((X - )/ < (0.314 - )/)

= P(Z < (0.314 - 0.958)/0.322)

= P(Z < -2)

= 0.0228

4) P(55.4 < X < 68.2)

= P((55.4 - )/ < (X - )/ < (68.2 - )/ )

= P((55.4 - 56.5)/7 < Z < (68.2 - 56.5)/7)

= P(-0.157 < Z < 1.671)

= P(Z < 1.671) - P(Z < -0.157)

= 0.9526 - 0.4376

= 0.515

5) P(X > 4)

= P((X - )/ > (4 - )/)

= PZ > (4 - 5.6)/2.1)

= P(Z < -0.762)

= 0.223