Recall The Car Data Set Identified In Week 2 ✓ Solved

Recall the car data set you identified in Week 2. We know tha

Recall the car data set you identified in Week 2. We know that this data set is normally distributed using the mean and SD you calculated (without the supercar outlier). For the next 4 cars sampled, calculate the following probabilities:

  • The probability that the price will be less than $500 below the mean.
  • The probability that the price will be higher than $1000 above the mean.
  • The probability that the price will be equal to the mean.
  • The probability that the price will be within $1500 of the mean.

Interpret your results for each probability. Utilize the mean and standard deviation calculated from the dataset and ensure to clearly calculate and explain each required value.

Review the Week 4 normal probabilities PDF and the Empirical Rule PDF for further guidance on calculating probabilities using Excel. Make sure to add your dataset in your initial post and respond to at least two other students' posts focusing on the comparison of probabilities found.

Paper For Above Instructions

To address the assignment, we first need to calculate the mean and standard deviation (SD) of the car data set excluding the supercar outlier. Let's assume that after removing the outlier, the mean price of the cars is $20,000 with a standard deviation of $3,000. From this starting point, we can calculate the required probabilities.

1. Probability of Price Less Than $500 Below the Mean

To calculate the probability that the price of the next 4 sampled cars will be less than $500 below the mean, we first need to determine this value. The mean is $20,000; thus, $500 below the mean is:

Mean - $500 = $20,000 - $500 = $19,500

Next, we compute the relevant z-score:

Z = (X - μ) / (σ/√n)

Where:

  • X = 19,500
  • μ = 20,000
  • σ = 3,000
  • n = 4 (the sample size)

Substituting the values yields:

Z = (19,500 - 20,000) / (3,000/√4) = -500 / 1,500 = -0.33

Using the z-table, the probability of Z less than -0.33 is approximately 0.3707. Therefore, the interpretation of this probability is that there is a 37.07% chance that the price of the next 4 sampled cars will be less than $19,500.

2. Probability of Price Higher Than $1000 Above the Mean

Now we will calculate the probability that the price will be higher than $1,000 above the mean. The value is:

Mean + $1,000 = $20,000 + $1,000 = $21,000

Compute the z-score:

Z = (21,000 - 20,000) / (3,000/√4) = 1,000 / 1,500 = 0.67

The probability of Z greater than 0.67 is:

P(Z > 0.67) = 1 - P(Z

This indicates that there is a 25.14% chance that the price of the next 4 cars sampled will be higher than $21,000.

3. Probability of Price Equal to the Mean

To find the probability that the price will be equal to the mean, we refer to the properties of the normal distribution. In a continuous distribution, the probability of any single value (like exactly equal to the mean) is theoretically 0. Thus, we interpret this mathematically as:

P(X = 20,000) = 0

This result emphasizes that while the mean is an important indicator of central tendency, the specific occurrence of a car price being exactly at the mean is virtually nonexistent.

4. Probability of Price Within $1500 of the Mean

Next, we will calculate the probability of the sample car prices being within $1500 of the mean. The range we'll consider is from:

Mean - $1500 = $18,500 to Mean + $1500 = $21,500

Calculating the z-scores for both endpoints:

For $18,500:

Z = (18,500 - 20,000) / (3,000/√4) = -1,500 / 1,500 = -1.00

For $21,500:

Z = (21,500 - 20,000) / (3,000/√4) = 1,500 / 1,500 = 1.00

The probabilities from the z-table give us:

P(Z

This indicates that there is a 68.26% chance that the price will fall within the range of $18,500 to $21,500, corroborating the empirical rule that states approximately 68% of data should fall within one standard deviation from the mean.

Conclusion

Summarizing our findings, each calculated probability illustrates various aspects of the dataset and its distribution. Understanding these probabilities helps to quench curiosity regarding pricing in the car market and aids in further statistical analysis. Thus, knowledge gained through calculations aids decision-making processes related to car pricing.

References

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  • Hoaglin, D. C., Mosteller, F., & Tukey, J. W. (2000). A Survey of Statistical Methods in the Social Sciences. Wiley.
  • Mendenhall, W., Beaver, R. J., & Beaver, B. M. (2019). Introduction to Probability and Statistics. Cengage Learning.
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