Week 5 Assignment Application of Normal Distribution

Question

In a survey of men aged 20-29 in a country, the mean height was 73.4 inches with a standard deviation of 2.7 inches. Find the minimum height in the top 10% of heights.

Here, the mean, μ, is 73.4 and the standard deviation, σ, is 2.7. Let x be the minimum height in the top 10% of heights. The area to the right of x is 10%=0.10. So, the area to the left of x is 1−0.10=0.90. Use Excel to find x.

Question

Two thousand students took an exam. The scores on the exam have an approximate normal distribution with a mean of μ=81 points and a standard deviation of σ=4 points. The middle 50% of the exam scores are between what two values?

Question

On average, 28 percent of 18 to 34 year olds check their social media profiles before getting out of bed in the morning. Suppose this percentage follows a normal distribution with a random variable X, which has a standard deviation of five percent. Find the probability that the percent of 18 to 34 year olds who check social media before getting out of bed in the morning is, at most, 32.

Question

A worn, poorly set-up machine is observed to produce components whose length X follows a normal distribution with mean 14 centimeters and variance 9. Calculate the probability that a component is at least 12 centimeters long.

Question

Sugar canes have lengths, X, that are normally distributed with mean 365.45 centimeters and standard deviation 4.9 centimeters. What is the probability of the length of a randomly selected cane being between 360 and 370 centimeters?

Question

Suppose that the weight, X, in pounds, of a 40-year-old man is a normal random variable with mean 147 and standard deviation 16. Calculate P(120≤X≤153).

Question

A tire company finds the lifespan for one brand of its tires is normally distributed with a mean of 47,500 miles and a standard deviation of 3,000 miles. What mileage would correspond to the the highest 3% of the tires?

Question

A firm’s marketing manager believes that total sales for next year will follow the normal distribution, with a mean of $3.2 million and a standard deviation of $250,000. Determine the sales level that has only a 3% chance of being exceeded next year.

Solution:

Question

In a survey of men aged 20-29 in a country, the mean height was 73.4 inches with a standard deviation of 2.7 inches. Find the minimum height in the top 10% of heights.

Open Excel. Click on an empty cell. Type =NORM.INV(0.90,73.4,2.7)and press ENTER.

The answer, rounded to one decimal place, is x≈76.9. Therefore, the minimum height in the top 10% of heights is 76.9 inches.

Question

The probability to the left of x1 is 0.25. Use Excel to find x1.

Open Excel. Click on an empty cell. Type =NORM.INV(0.25,81,4)and press ENTER.

Rounding to the nearest integer, x1≈78. The probability to the left of x2 is 0.25+0.50=0.75. Use Excel to find x2.

Open Excel. Click on an empty cell. Type =NORM.INV(0.75,81,4)and press ENTER.

Rounding to the nearest integer, x2≈84. Thus, the middle 50% of the exam scores are between 78 and 84.

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