W4 Midterm Business Statistics: There Are 8 Questions ✓ Solved

W4 Midterm Business Statistics Midterm There are 8 questions

Question 1 The following shows the temperatures (high, low) and weather conditions in a given Sunday for some selected world cities. For the weather conditions, the following notations are used: c = clear; cl = cloudy; sh = showers; pc = partly cloudy.

1. How many elements are in this data set?

2. How many variables are in this data set?

3. How many observations are in this data set?

4. Name the variables and indicate whether they are categorical or quantitative.

Question 2 A student has completed 20 courses in the School of Arts and Sciences. Her grades in the 20 courses are shown below.

5. Develop a frequency distribution and a bar chart for her grades.

6. Develop a relative frequency distribution for her grades and construct a pie chart.

Question 3 The number of hours worked per week for a sample of ten students is shown below.

7. Determine the median and explain its meaning.

8. Compute the 70th percentile and explain its meaning.

9. What is the mode of the above data? What does it signify?

Question 4 You are given the following information on Events A, B, C, and D.

10. Compute P(D).

11. Compute P(A ∩ B).

12. Compute P(A | C).

13. Compute the probability of the complement of C.

14. Are A and B mutually exclusive? Explain your answer.

15. Are A and B independent? Explain your answer.

16. Are A and C mutually exclusive? Explain your answer.

17. Are A and C independent? Explain your answer.

Question 5 When a particular machine is functioning properly, 80% of the items produced are non-defective.

18. If three items are examined, what is the probability that one is defective?

19. Use the binomial probability function to answer this question.

Question 6 The average starting salary of this year’s graduates of a large university (LU) is $20,000 with a standard deviation of $8,000. Furthermore, it is known that the starting salaries are normally distributed.

20. What is the probability that a randomly selected LU graduate will have a starting salary of at least $30,400?

21. Individuals with starting salaries of less than $15,600 receive a low income tax break. What percentage of the graduates will receive the tax break?

22. What are the minimum and the maximum starting salaries of the middle 95.4% of the LU graduates?

Question 7 A simple random sample of 6 computer programmers in Houston, Texas revealed the sex of the programmers and the following information about their weekly incomes.

23. What is the point estimate for the average weekly income of all the computer programmers in Houston?

24. What is the point estimate for the standard deviation of the population?

25. Determine a point estimate for the proportion of all programmers in Houston who are female.

Question 8 Students of a large university spend an average of $5 a day on lunch. The standard deviation of the expenditure is $3. A simple random sample of 36 students is taken.

26. What are the expected value, standard deviation, and shape of the sampling distribution of the sample mean?

27. What is the probability that the sample mean will be at least $4?

28. What is the probability that the sample mean will be at least $5.90?

Paper For Above Instructions

This paper provides concise solutions, formulas, and interpretations for the eight midterm questions in business statistics. Where specific raw data are not provided in the prompt, I state the method and give numerical solutions for those items with sufficient numeric information (Questions 5, 6, and 8). For conceptual problems and unspecified data I present the appropriate formulas and interpretation steps so answers can be completed when the raw data are available (Triola, 2018; Devore, 2015).

Question 1: Elements, variables, observations, and types

Definitions: An element (or unit) is each entity described by the data set; a variable is any measured attribute; an observation is the set of values for all variables for a single element (Wackerly, Mendenhall, & Scheaffer, 2008).

Application to the weather dataset: Each city listed is an element. If the dataset lists, for each city, a high temperature, a low temperature, and a weather code (c, cl, sh, pc), then the number of variables is three (high temp, low temp, weather condition). The number of observations equals the number of elements (cities). Variable types: high temperature and low temperature are quantitative (continuous or interval/ratio scale); weather condition is categorical (nominal) (Triola, 2018).

Question 2: Frequency and relative frequency distributions and charts

Procedure: Given 20 course grades, first list unique grade categories (e.g., A, B, C, D, F or numeric bins). Construct a frequency table giving counts for each category and then compute relative frequency = frequency / 20 (Moore, 2017). For a bar chart, plot categories on the x-axis and frequencies on the y-axis. For the pie chart, use relative frequencies as slice proportions.

Example method: If grades are numeric, choose class intervals (equal width), tally counts, and draw the bar chart. If letter grades, count occurrences of each letter and draw bars and pie slices. This approach conforms with standard practice for categorical and quantitative summarization (Triola, 2018).

Question 3: Median, percentile, and mode

Median: Sort the 10-hour values in ascending order; if n is even (n=10) the median is the average of the 5th and 6th values. Interpretation: the median is the 50th percentile—half the sampled students work at most that number of hours and half work at least that number (Helsel & Hirsch, 2002).

70th percentile: Sort data and locate position k = (70/100)(n+1) = 0.711 = 7.7, interpolate between 7th and 8th ordered values (if needed) to compute the 70th percentile. Interpretation: 70% of students work at most that many hours (or, equivalently, 30% work more).

Mode: The mode is the most frequently occurring value(s). Its significance: identifies the most common number of hours worked in the sample; mode is most useful for categorical or discrete numeric data (Bluman, 2014).

Question 4: Event probabilities, complements, mutual exclusivity, and independence

General formulas and steps (Ross, 2014):

• P(complement of C) = 1 − P(C).
• P(A ∩ B) is computed directly if joint probability is given or via P(A)P(B) if independent; otherwise use P(A ∩ B) = P(A) + P(B) − P(A ∪ B) if union known.
• P(A | C) = P(A ∩ C) / P(C), provided P(C) > 0.
• Mutually exclusive: A and B are mutually exclusive if P(A ∩ B) = 0. Independence: A and B are independent if P(A ∩ B) = P(A)P(B).

Application: compute P(D), P(A ∩ B), and conditional probabilities using the provided numeric probabilities when they are available; test mutual exclusivity via intersection = 0 and independence via product rule (Ross, 2014).

Question 5: Binomial probability for defective items

Given: non-defective rate = 0.80, so defective p = 0.20; n = 3. The probability exactly k defects in n trials follows the binomial probability:

P(X = k) = C(n, k) p^k (1 − p)^(n−k) (Devore, 2015).

Compute probability exactly one defective (k = 1): P(X = 1) = C(3,1)(0.2)^1(0.8)^2 = 3 0.2 0.64 = 0.384. Interpretation: there is a 38.4% chance that exactly one of three items is defective when the defect rate is 20% (Devore, 2015).

Question 6: Normal distribution computations

Given: μ = 20,000; σ = 8,000; normality assumed.

a) P(X ≥ 30,400): z = (30,400 − 20,000) / 8,000 = 1.30. From standard normal tables, P(Z ≥ 1.30) ≈ 1 − 0.9032 = 0.0968 (≈ 9.68%) (Wackerly et al., 2008).

b) P(X

c) Middle 95.4% for a normal distribution corresponds approximately to ±2σ. Hence lower bound = μ − 2σ = 20,000 − 16,000 = 4,000; upper bound = μ + 2σ = 20,000 + 16,000 = 36,000 (Devore, 2015).

Question 7: Point estimates from a sample

Given a simple random sample of size n = 6 with weekly incomes and sexes recorded:

• Point estimate for the population mean weekly income: the sample mean x̄ = (1/n) Σ xi (unbiased estimator of μ for simple random sampling) (Moore, 2017).
• Point estimate for the population standard deviation: the sample standard deviation s = sqrt[(1/(n−1)) Σ (xi − x̄)^2] is the standard estimator for σ when population data are unknown (Wackerly et al., 2008).
• Point estimate for proportion female: p̂ = (number of females in sample) / n. This is an unbiased point estimate for the population proportion p (Agresti & Franklin, 2013).

Interpretation: compute these using the six observed incomes and observed female count; report appropriate units and sampling assumptions (random sampling) (Moore, 2017).

Question 8: Sampling distribution of the sample mean and probabilities

Given population mean μ = $5, population σ = $3, sample size n = 36. By the Central Limit Theorem the sampling distribution of the sample mean x̄ has:

• Expected value E(x̄) = μ = $5.
• Standard deviation (standard error) = σ / √n = 3 / 6 = $0.50.
• Shape: approximately normal (exactly normal if original distribution is normal; approximately normal for large n by CLT) (Casella & Berger, 2002).

a) P(x̄ ≥ 4): z = (4 − 5) / 0.5 = −2. P(x̄ ≥ 4) = P(Z ≥ −2) = Φ(2) ≈ 0.9772 (≈97.72%) (Triola, 2018).

b) P(x̄ ≥ 5.90): z = (5.90 − 5) / 0.5 = 1.8. P(x̄ ≥ 5.90) = 1 − Φ(1.8) ≈ 1 − 0.9641 = 0.0359 (≈3.59%).

Summary interpretation: For the items where numeric data were provided (Questions 5, 6, 8) we computed exact probabilities. For dataset-dependent items (Questions 1–4 and 7) the solution provides the correct definitions, formulas, and steps to compute the requested statistics once the raw data or event probabilities are supplied. These methods align with standard introductory statistical practice and tables referenced below (Devore, 2015; Triola, 2018; Ross, 2014).

References

• Agresti, A., & Franklin, C. (2013). Statistics: The Art and Science of Learning from Data. Pearson Education.
• Bluman, A. G. (2014). Elementary Statistics: A Step by Step Approach. McGraw-Hill Education.
• Casella, G., & Berger, R. L. (2002). Statistical Inference (2nd ed.). Duxbury.
• Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences (9th ed.). Cengage Learning.
• Moore, D. S., Notz, W. I., & Fligner, M. A. (2017). The Basic Practice of Statistics (7th ed.). W. H. Freeman.
• Ross, S. M. (2014). A First Course in Probability (9th ed.). Pearson.
• Triola, M. F. (2018). Elementary Statistics (13th ed.). Pearson.
• Wackerly, D., Mendenhall, W., & Scheaffer, R. (2008). Mathematical Statistics with Applications (7th ed.). Brooks/Cole.
• National Institute of Standards and Technology (NIST) Engineering Statistics Handbook. (2020). https://www.itl.nist.gov/div898/handbook/
• StatTrek. (2019). Statistics and Probability Tutorials and Tools. https://stattrek.com/