Midterm Business/Math Statistics

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Mathematics

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See attached Word document for the list of eight questions with graphic.

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Midterm There are 8 questions in total, each worth 12.5 points. Please upload your answers to the dropbox below. 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. 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. 1. 2. 3. 4. How many elements are in this data set? How many variables are in this data set? How many observations are in this data set? 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. 1. 2. Develop a frequency distribution and a bar chart for her grades. 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. 1. 2. 3. Determine the median and explain its meaning. Compute the 70th percentile and explain its meaning. 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. 1. 2. 3. 4. 5. Compute P(D). Compute P(A ∩ B). Compute P(A | C). Compute the probability of the complement of C. Are A and B mutually exclusive? Explain your answer. 6. 7. 8. 1. 2. 3. Are A and B independent? Explain your answer. Are A and C mutually exclusive? Explain your answer. 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. If three items are examined, what is the probability that one is defective? 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. 1. 2. 3. What is the probability that a randomly selected LU graduate will have a starting salary of at least $30,400? 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? 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. 1. 2. 3. What is the point estimate for the average weekly income of all the computer programmers in Houston? What is the point estimate for the standard deviation of the population? 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. 1. 2. What are the expected value, standard deviation, and shape of the sampling distribution of the sample mean? What is the probability that the sample mean will be at least $4? 3. What is the probability that the sample mean will be at least $5.90?
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Explanation & Answer

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Question 1
i.
ii.
iii.
iv.

There are 28 elements in the data set.
There are 4 variables in the data set.
There are 7 observations in the data set.
The variables and their type are:
Variable
Type
City
Categorical
High temperature (Hi)
Quantitative
Low temperature (Lo)
Quantitative
Condition
Categorical

Question 2
i.

The frequency distribution and bar chart of the grades are presented below:
Grade
A
B
C
Total

ii.

Frequency
4
11
5
20

The relative frequency distribution of the grades and the corresponding pie chart are
presented below:
Variable
A
B
C
Total

Relative Freq.
4
= 0.20
20
11
= 0.55
20
5
= 0.25
20
1.00

Question 3
The number of hours worked per week sorted in ascending order is presented below:
0, 6, 8, 16, 18, 20, 22, 30, 40, 40
18+20

= 19 hours worked per week

i.

The median is:

ii.

𝑀𝑒𝑑𝑖𝑎𝑛 = 19
The median value means that 50% of the students work 19 hours or less per week.
Alternatively, the median can be interpreted as 50% of the students work 19 hours or
more per week.
𝑖𝑛𝑑𝑒𝑥 = 0.7 × 10 = 7

2

70th percentile, 𝑃70 =

7𝑡ℎ 𝑣𝑎𝑙𝑢𝑒+8𝑡ℎ 𝑣𝑎𝑙𝑢𝑒
2

=

22+30
2

= 26

𝑃70 = 26
The 70th percentile means that 70% of the students worked less than 26 hours per week.
iii.

The mode of the data is 40. The mode signifies the most frequent number of hours
worked per week by the students.

Question 4
Given: P(A) = 0.4,
P(A ∩ C) = 0.04

P(B) = 0.2,
P(A ∪ D) = 0.6

i.

P(A ∪ D) = P(A) + P(D) − P(A ∩ D)
∴ P(D) = P(A ∪ D) − P(A) + P(A ∩ D)
P(D) = 0.6 − 0.4 + 0.03
P(D) = 0.23

ii.

𝑃(𝐴|𝐵) =

𝑃(𝐴∩𝐵)
𝑃(𝐵)

∴ 𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴|𝐵) ∙ 𝑃(𝐵)
𝑃(𝐴 ∩ 𝐵) = 0.3 × 0.2
𝑃(𝐴 ∩ 𝐵) = 0.06
iii.

𝑃(𝐴|𝐶) =

𝑃(𝐴∩𝐶)
𝑃(𝐶)

=

0.04
0.1...


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