Empirical Rule & Chebyshevs

1. Calorie Intake and Z-scores

A. Z-score for Teen with 2200 calories:

1. Calculate the difference between the individual calorie intake (2200) and the average intake for Teens (1700).

Difference (X - μ) = 2200 calories - 1700 calories = 500 calories

2. Divide the difference by the standard deviation for Teens (400 calories).

Z-score = (X - μ) / σ = 500 calories / 400 calories = 1.25

B. Similar Calorie Intake for PreTeen:

1. Find the z-score for the Teen with 1900 calories (follow steps from 1A).

2. Since we know the Teen's z-score and want to find the corresponding calorie intake for a PreTeen, we need to use the z-score and standard deviation for PreTeens (350 calories).

3. The z-score represents the number of standard deviations a specific value is away from the mean. We can use the following formula to find the corresponding value in the PreTeen data set:

PreTeen Calorie Intake = μ (PreTeen) + (z-score) * σ (PreTeen)

where:

• μ (PreTeen) = Average calorie intake for PreTeens (1500 calories)
• σ (PreTeen) = Standard deviation for PreTeens (350 calories)
• z-score = Z-score from Teen with 1900 calories (calculated in step 1B.1)

2. Travel Time on 4th of July

A. Probability of Traveling More Than 57 Minutes:

1. The average travel time is 68 minutes, and the standard deviation is 11 minutes. We know the empirical rule (or the 68-95-99.7 rule) states that 68% of the data falls within 1 standard deviation of the mean.

2. To find the probability of traveling more than 57 minutes, we need to consider the area to the right of 57 minutes. Since the normal distribution is symmetrical, half (34%) of the data falls within 1 standard deviation above the mean (68 minutes + 11 minutes = 79 minutes).

3. There's another 34% of the data that falls beyond 79 minutes (to the right), but we only care about the portion exceeding 57 minutes. Unfortunately, the empirical rule doesn't provide exact percentages beyond 1 standard deviation.

B. Probability of Traveling At Most 90 Minutes:

1. Similar to question 2A, we consider the area to the left of 90 minutes, which includes the data within 1 standard deviation (68%) and potentially some data beyond that.

2. Following the empirical rule, 68% of the data falls within 1 standard deviation (between 57 minutes and 79 minutes).

3. There's an additional 16% (half of the 34%) that falls between 79 minutes and 90 minutes (1 standard deviation above the mean).

C. Probability of Traveling Between 35 and 46 Minutes:

1. One standard deviation below the mean is 68 minutes - 11 minutes = 57 minutes.

2. Two standard deviations below the mean is 57 minutes - 11 minutes = 46 minutes.

3. The empirical rule states that 68% of the data falls within 1 standard deviation of the mean. We only care about half (34%) of that data, which falls between 46 minutes and 57 minutes.

3. Minimum Proportion Traveling Between 36 and 90 Miles

1. We know the average travel distance is 63 miles and the standard deviation is 14 miles. One standard deviation below the mean is 63 miles - 14 miles = 49 miles.

2. One standard deviation above the mean is 63 miles + 14 miles = 77 miles.

3. Two standard deviations below the mean is 49 miles - 14 miles = 35 miles.

4. Two standard deviations above the mean is 77 miles + 14 miles = 91 miles.

5. We want the minimum proportion that travels between 36 miles and 90 miles. The empirical rule only provides estimates for 1, 2

Sample Solution