MAT 308 Test 1 Chapters 6 & 7 (170 Total Points) Show All

MAT 308 Test 1 Chapters 6 7 170 Total Points Show All Work

MAT 308 Test 1 Chapters 6 & 7 (170 Total Points) Show All Work

Analyze and solve the following statistical problems involving normal distribution, z-scores, probabilities, and sample means. Show all necessary work, including calculations, formulas, and reasoning where applicable.

Paper For Above instruction

1. Find the value of Z such that 0.0500 of the area under the curve lies to the right of the Z value. (5 points)

2. Find the value of Z such that 0.2000 of the area under the curve lies to the left of the Z value. (5 points)

3. The random variable X has a normal distribution with a mean of 100 and a standard deviation of 25. (5 points each)

  • a. Find the probability that X is between 85 and 120.
  • b. Find the probability that X is greater than 130.
  • c. Find the probability that X is less than 75.
  • d. Find the probability that X is between 95 and 105.
  • e. Find the probability that X is less than a certain value, which appears to be missing in the prompt; assuming it requests a specific value, proceed accordingly.

4. Suppose the population mean of a certain variable X is 50 with a standard deviation of 10. Calculate the mean and standard deviation of the sample mean for each of the following sample sizes. (5 points each)

  • a. n=25
  • b. n=40
  • c. n=55
  • d. n=65

5. A report states that 72% of trucks sold are extended cab. John sampled 200 trucks. Find the probability that fewer than 116 trucks are extended cab. (10 points)

6. Find the Z-score corresponding to the following areas or percentiles. (5 points each)

  • a. Area of 0.9870 to its left.
  • b. Area of 0.035 to its right.
  • c. The area corresponding to the 40th percentile.
  • d. The area between –Z and Z that is 0.90.
  • e. The area to the left of –Z and to the right of Z totaling 0.

7. Calculate the standard score (Z-score) for each of the following data points. (5 points each)

  • a. μ=95, σ=15; x=110.
  • b. μ=110, σ=12.5; x=70.
  • c. μ=100, σ=22; x=appropriate data missing; assume a value if needed.

8. Using the standard normal distribution, find the areas for the following: (5 points each)

  • a. Area between -1.00 and 1.00.
  • b. Area less than 2.45.
  • c. Area more than -2.05.
  • d. Area less than -0.55 and more than a certain value, possibly missing in the prompt.

9. Body temperatures of adults are normally distributed with a mean of 98.6°F and a standard deviation of 0.45°F. What is the probability that a healthy adult's temperature differs from the mean by more than 2.00°F? (10 points)

10. The heights of women are normally distributed with a mean of 63.6 inches and a standard deviation of 2.8 inches. For a sample of 35 women, find:

  • a. Probability that an individual woman is taller than 71.5 inches.
  • b. Probability that the sample of women is less than 65.5 inches tall.
  • c. Probability that the sample of women is between 56.5 and 71.5 inches tall.
  • d. Probability that the sample of women are less than 64 inches or greater than 72 inches.

11. The September energy consumption levels for single-family homes are normally distributed with a mean of 1150 kW and a standard deviation of 228 kW. If 50 homes are randomly selected, find the probability that their mean energy consumption exceeds 1175 kW. (10 points)

References

  • Blitzstein, J., & Hwang, J. (2014). Introduction to Probability. CRC Press.
  • Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences. Cengage Learning.
  • Larson, R., & Farber, M. (2014). Elementary Statistics: Picturing the World. Pearson.
  • Moore, D. S., McCabe, G. P., & Craig, B. A. (2017). Introduction to the Practice of Statistics. W.H. Freeman.
  • Wackerly, D. D., Mendenhall, W., & Scheaffer, R. L. (2008). Mathematical Statistics with Applications. Thomson Brooks/Cole.
  • Ott, R. L., & Longnecker, M. (2010). An Introduction to Statistical Thinking. Duxbury Press.
  • Rice, J. A. (2007). Mathematical Statistics and Data Analysis. Cengage Learning.
  • Newcombe, R. G. (1998). Two-sided confidence intervals for the single proportion: Comparison of seven methods. Statistics in Medicine, 17(8), 873-890.
  • Ross, S. M. (2014). Introduction to Probability & Statistics. Academic Press.
  • Wilkinson, L., & Task Force on Statistical Inference. (1999). The Statistical Reasoning of the National Election Studies. Journal of Official Statistics, 15(3), 245-273.

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