Statistics Sample Variance Using Three Different Methods Questions

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QNA23

Mathematics

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Name: ____________________________________________ Date:___________________ Quiz 1 – Intro to Probability & Statistics Directions: Answer each question to the best of your ability. Please show all your work. This quiz is due June 30th by 11:50 pm. 1. Use the following dataset to answer parts a-f. 𝑥: 7, 82, 101, 89, 99, 104, 79, 95 a. Find the sample mean. b. Find the sample standard deviation and provide a short interpretation of the value. c. Find the sample variance in 3 different ways. d. Find the sample median. e. Are there any outliers present in the data? If so, what value(s)? f. Construct a stem and leaf plot. 2. Fifteen WVU students were polled to ask the number of siblings each student had. The results of the poll are listed below: 𝑥: 0, 4, 2, 2, 1, 3, 0, 1, 1, 6, 9, 2, 1, 3, 2 a. Is the data widely dispersed? Please explain your reasoning. b. Find the sample median. c. Find the IQR. d. Construct a boxplot. e. Construct a dot plot 3. Sketch a histogram that is symmetric and unimodal. 4. A box contains 24 batteries of which four are defective. Two batteries are selected at random and tested. Let A be the event that the first battery is defective. Let B be the event that the second battery drawn in defective. a. Find P(A) b. Find P(B|A) c. Find P(A ∩ B) d. Find P(Ac ∩ B) e. Find P(B) f. Find P(A|B) g. Are A and B independent? Justify your answer 5. Consider the box problem in Number 4, where there exists a box of 24 batteries of which four are defective. If one person selects 10 batteries from the box, and a second person selects the remaining 14 batteries, what is the probability that all four defective batteries will be obtained by the same person? 6. Let 46 percent of the families in a certain city subscribe to the morning newspaper, 62 percent of the families subscribe to the afternoon newspaper, and 82 percent subscribe to at least one of the two newspapers. a. Find the probability that a family subscribes to both newspapers. b. Find the probability that a family subscribes to neither of the two newspapers.
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Explanation & Answer

Here is the complete work.

Solution
1. Given dataset is x: 7, 82, 101, 89, 99, 104, 79, 95
We can arrange the given data in ascending order as follows
x: 7, 79, 82, 89, 95, 99, 101, 104

n

∑ xi

a. Sample mean: x
⏨=

i=1

=

n

7 + 79 + 82 + 89 + 95 + 99 + 101 + 104
656
=
= 82
8
8

Ans. Hence, sample mean is 82.
b. Sample Standard deviatio
n

∑ (x - xi)2

i=1

s=

n-1

2

xi

xi - x⏨

xi - x⏨

7

7 - 82 = -75

5625

79

79 - 82 = -3

9

82

82 - 82 = 0

0

89

89 - 82 = 7

49

95

95 - 82 = 13

169

99

99 - 82 = 17

289

101

101 - 82 = 19

361

104

104 - 82 = 22

484

8


i=1

xi - x⏨ = 6986

n

Therefore, sample standard deviation, s =

∑ (x - xi)2

i=1

=

n-1

6986
=
8-1

6986
= 31.59
7

Hence, s = 31.59
Interpretation: Since s = 31.59, the data seems to be highly scattered around the mean.
c. Sample Variance using three different methods:

(i) From the table (shown above), sample variance s2 =

n

∑ (x - xi)2

i=1

n-1

=

(ii) Using the sample standard deviation.
In part b, we have calculated sample standard deviation s = 31.59
sample variance = s2 = 31.592 = 997.93 ≈ 998

(iii) Sample variance can be...


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