ProficientWriting | Analyzing Caloric Intake and Travel Times Using Statistical Concepts

Analyzing Caloric Intake and Travel Times Using Statistical Concepts

Discussion Board 3 Z-Score, Empirical Rule & Chebyshevs

The following data about caloric intake comes from the FDA:
Teen (T) PreTeen (PT)
1,700 Average Calories 1,500
400 Standard Dev. 350
A. If a Teen has a caloric intake of 2.200 calories, what would their z-score be? (No rounding, full answer)
B. If a Teen has a caloric intake of 1,900 calories, what would be a similar caloric intake for a PreTeen? (need to find a z-score for Teen and then use that information in computing for the PreTeen) (No rounding, full answer)
2.The average amount of time a person travels for the 4th of July holiday is 68 minutes with a standard deviation of 11 minutes.If the average amount of time a person travels for the 4th of July holiday is normally distributed, answer the following questions using the empirical rule (No rounding, full answer either answer acceptable .2236 or 22.36%):
2A. What is the probability that someone will travel more than 57 minutes for the holiday?
2B. What is the probability that someone will travel at most 90 minutes for the holiday?
2C. What is the probability that someone will travel between 35 to 46 minutes for the holiday? Chris OByrne 2024
The average number of miles someone travels for the 4th of July holiday is 63 miles with a standard deviation of 14 miles.What is the minimum proportion(percentage) of people that will travel between 36 miles and 90 miles? (Round only your final answer to 4 decimal places and express as a percentage ex. .6123 expressed as 61.23%) (do not round interim computations)

Analyzing Caloric Intake and Travel Times Using Statistical Concepts

1. Caloric Intake Data Analysis

Given data for caloric intake:

– Teen: Average Calories = 1,700, Standard Deviation = 400
– PreTeen: Average Calories = 1,500, Standard Deviation = 350

A. Z-Score Calculation for a Teen with 2,200 Calories:

Z-Score formula: Z = (X – μ) / σ

For a Teen with 2,200 calories:
Z = (2,200 – 1,700) / 400 = 0.75

B. Finding Similar Caloric Intake for a PreTeen with 1,900 Calories:

To find a similar caloric intake for a PreTeen, we need to calculate the z-score for the Teen first:

Z for 1,900 calories for Teen: (1,900 – 1,700) / 400 = 0.5

Now, using the z-score of 0.5 for the Teen, we can calculate the similar caloric intake for a PreTeen:

Similar Caloric Intake for PreTeen = (0.5 * 350) + 1,500 = 1,675 calories

2. Travel Time Analysis Using Empirical Rule

Given data for travel time:

– Average Time = 68 minutes, Standard Deviation = 11 minutes

A. Probability of Traveling More Than 57 Minutes:

Z-Score for 57 minutes: (57 – 68) / 11 = -1

Using the empirical rule:
Probability of traveling more than 57 minutes = P(Z > -1) = 1 – P(Z ≤ -1) = 1 – 0.1587 = 0.8413 or 84.13%

B. Probability of Traveling at Most 90 Minutes:

Z-Score for 90 minutes: (90 – 68) / 11 = 2

Probability of traveling at most 90 minutes = P(Z ≤ 2) = 0.9772 or 97.72%

C. Probability of Traveling Between 35 to 46 Minutes:

Z-Score for 35 minutes: (35 – 68) / 11 ≈ -3
Z-Score for 46 minutes: (46 – 68) / 11 ≈ -2

Using the empirical rule:
P(-3 < Z < -2) ≈ P(Z < -2) – P(Z < -3) ≈ 0.0228 or 2.28%

3. Proportion of People Traveling Between 36 Miles and 90 Miles

Given data for miles traveled:

– Average Miles = 63 miles, Standard Deviation = 14 miles

Calculating Z-Scores:
Z for 36 miles: (36 – 63) / 14 ≈ -1.93
Z for 90 miles: (90 – 63) / 14 ≈ +1.93

Using the Chebyshev’s theorem:
Minimum proportion between 36 to 90 miles is at least:

1 – (1 / (1^2)) ≈ 0.75 or 75%

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