High School

Toby's Trucking Company:

1. The distance traveled per truck per year is normally distributed, with a mean of 50 thousand miles and a standard deviation of 12 thousand miles.
- a. What proportion of trucks can be expected to travel between 34 thousand and 50 thousand miles in a year?
- b. What percentage of trucks can be expected to travel either less than 30 thousand or more than 60 thousand miles in a year?
- c. How many miles will be traveled by at least 80 percent of the trucks?
- d. What would be your answers to (a) through (c) if the standard deviation were 10 thousand miles?

2. In 2017, the per capita consumption of bottled water in China was reported to be 25.46 gallons. Assume that the per capita consumption of bottled water in China is approximately normally distributed with a mean of 25.46 gallons and a standard deviation of 8 gallons.
- a. What is the probability that someone in China consumed more than 33 gallons of bottled water in 2017?
- b. What is the probability that someone in China consumed between 10 and 20 gallons of bottled water in 2017?
- c. What is the probability that someone in China consumed less than 10 gallons of bottled water in 2017?
- d. Ninety-nine percent of the people in China consumed less than how many gallons of bottled water?

3. The fill amount in 2-liter soft drink bottles is normally distributed, with a mean of 2.0 liters and a standard deviation of 0.05 liters. Bottles with less than 95% of the listed net content (1.90 liters) may result in penalties, and bottles above 2.10 liters may cause spillage.
- a. What proportion of the bottles will contain between 1.90 and 2.0 liters?
- b. What proportion of the bottles will contain between 1.90 and 2.10 liters?
- c. What proportion of the bottles will contain below 1.90 liters or above 2.10 liters?
- d. At least how much soft drink is contained in 99% of the bottles?
- e. Ninety-nine percent of the bottles contain an amount that is between which two values (symmetrically distributed) around the mean?

4. Interns report important factors when deciding where to work include career growth, salary and compensation, location and commute, and company culture and values. According to Glassdoor, the mean monthly pay of interns at Intel is $5,940, with a standard deviation of $400.
- a. What is the probability that the monthly pay of an intern at Intel is less than $5,900?
- b. What is the probability that the monthly pay is between $5,700 and $6,100?
- c. What is the probability that the monthly pay is above $6,500?
- d. Ninety-nine percent of the intern monthly pays are higher than what value?
- e. Ninety-five percent of the intern monthly pays are between what two values, symmetrically distributed around the mean?

Answer :

Final answer:

a. The proportion of trucks expected to travel between 34,000 and 50,000 miles is approximately 40.82%. b. The percentage of trucks expected to travel less than 30,000 or more than 60,000 miles is approximately 26.6%. c. At least 80% of the trucks will travel at least 60,080 miles.

Explanation:

a. To find the proportion of trucks expected to travel between 34,000 and 50,000 miles, we need to calculate the z-scores for these values using the formula z = (x - μ) / σ where x is the value, μ is the mean, and σ is the standard deviation. The z-score for 34,000 miles is z = (34,000 - 50,000) / 12,000 = -1.33 and the z-score for 50,000 miles is z = (50,000 - 50,000) / 12,000 = 0. To find the proportion between these two values, we can use a standard normal distribution table. The area under the curve between -1.33 and 0 is approximately 0.4082 or 40.82%.

b. To find the percentage of trucks expected to travel less than 30,000 or more than 60,000 miles, we need to calculate the z-scores for these values. The z-score for 30,000 miles is z = (30,000 - 50,000) / 12,000 = -1.67 and the z-score for 60,000 miles is z = (60,000 - 50,000) / 12,000 = 0.83. To find the proportion outside these two values, we can subtract the area under the curve between -1.67 and 0.83 from 1. The area under the curve between -1.67 and 0.83 is approximately 0.734, so the percentage outside these two values is 1 - 0.734 = 0.266 or 26.6%.

c. To find the number of miles traveled by at least 80% of the trucks, we need to find the z-score that corresponds to the 80th percentile. We can find this value using a standard normal distribution table or a z-table. The z-score corresponding to the 80th percentile is approximately 0.84. We can then use the formula z = (x - μ) / σ to find the value of x. Substituting 0.84 for z, 0.84 = (x - 50,000) / 12,000. Solving for x, we get x = 0.84 * 12,000 + 50,000 = 60,080 miles.

d. If the standard deviation were 10,000 miles instead of 12,000 miles, the calculations for parts a, b, and c would be the same, except we would use the new standard deviation value in the calculations.

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