Central Limit Theorem and the Normal Distribution

Central Limit Theorem

Start by reviewing the following Handy Helper, which is also provided in this week’s Lesson: Week 4 Normal Probabilities.pdf. This document will give you step by step examples on how to use Excel to calculate probabilities using the Central Limit Theorem and the Normal Distribution. DO NOT do these calculations by hand. Let Excel do the heavy lifting for you.

Next, use the data identified in Car research excel, and find the mean and standard deviation for your car prices. Once you find your mean and standard deviation, assume those are the population mean, μ, and population standard deviation, σ. Next assume that car prices are normally distributed. Solve the following:

1. If you were to find another random sample of 4 car prices, what is the probability that the sample mean, X̅, of the new 4 cars will be less than$500 below the mean of your original data? Interpret your results.
   • HINT: we need to calculate the standard error when the sample size is 4 (called new sd in the Handy Helper provided in the first paragraph of this forum description). Additionally, we need to find the value that is $500 below the mean of your original data. For example, if your original mean for the 10 cars you observed in week 1 was $15,000, then $500 below the mean would be $14,500. Thus, the probability you want to find is P(X̅ < 14,500)
2. If you were to find another random sample of 4 car prices, what is the probability that the sample mean, X̅, of the new 4 cars will be higher than$1,000 above the mean of your original data? Interpret your results.
   • HINT: Use the same logic as above. If the mean of your original data is $15,000, then $1,000 above is 15,000 + 1,000 = $16,000. Thus the probability you would want to find is P(X̅ > 16,000).
3. If you were to find another random sample of 4 car prices, what is the probability that the sample mean ,X̅ , of the new 4 cars will be within$1,500 of the mean of your original data? Interpret your results.
   • HINT: Use the same logic as above. If the mean of your original data is $15,000, then $1,500 within your mean is 15,000 – 1,500 = $13,500 and 15,000 + 1,500 = $16,500. Thus the probability you would want to find is P(13,500 < X̅ < 16,500).

BONUS QUESTION: If you were to randomly sample 1 additional car, what is the probability that the new car price, X, will be at least $51,000? Thus, find the following probability: P(X ≥ 51,000)

The post should be at least 150 words. Also, attach the Excel file with the formulas that you used to solve the questions above. The bonus question is recommended (you not lose points for an incorrect solution).