Probability and Expected Value

This document discusses the concept of expected value for a random variable. It explains how to calculate the expected value for a discrete random variable and provides examples. The document also covers probability calculations and variance. Suitable for students studying probability and statistics.

QUESTION 1 Probability
(a) The expected value is a measure of central location(distribution) for a random
variable. It measures the population mean for a random variable.
The expected value for a discrete random variable is obtained by adding up the
product of the value of the random variable and its associated probability, taken over
all the values of the random variable (Newbold, Carlson & Thorne, 2013). That is,
.
For example, if we roll a normal six-sided die, the probability of landing on
1,2,3,4,5,or 6 is 1/6. To find the expected value when we roll a fair die we have:
(b) 1.
2. What were the average daily sales? 2.9 units
3. What was the probability of selling 2 or more loaves on any one day? 0.85
4. What was the probability of selling 3 or less? 0.65
5. What is the variance of the distribution? 6.67
6. What is the standard deviation? 2.58
Sales units
(x)
Number of
days (f) P(x) Exp Value
E(x) More than Less than [x-E(x)]^2 [x-E(x)]^2 P(x)
0 5 0.05 0 0.95 0.05 0 0
1 10 0.10 0.1 0.85 0.15 0.81 0.081
2 25 0.25 0.5 0.6 0.4 2.25 0.5625
3 25 0.25 0.75 0.35 0.65 5.0625 1.265625
4 20 0.20 0.8 0.15 0.85 10.24 2.048
5 15 0.15 0.75 0 1 18.0625 2.709375
Total 100 1.00 2.90 Variance 6.67
Average

daily sales
Total sales 290
Total days 100
Average 2.9
Standard
Deviation 2.58
Sales
units (x)
Number
of days
(f)
P(x)
Exp Value
E(x) More
than
Less
than [x-E(x)]^2 [x-E(x)]^2
P(x)
0 5 =C3/$C$9 =B3*D3 =1-D3 =D3 =(B3-E3)^2 =H3*D3
1 10 =C4/$C$9 =B4*D4 =F3-D4 =G3+D4 =(B4-E4)^2 =H4*D4
2 25 =C5/$C$9 =B5*D5 =F4-D5 =G4+D5 =(B5-E5)^2 =H5*D5
3 25 =C6/$C$9 =B6*D6 =F5-D6 =G5+D6 =(B6-E6)^2 =H6*D6
4 20 =C7/$C$9 =B7*D7 =F6-D7 =G6+D7 =(B7-E7)^2 =H7*D7
5 15 =C8/$C$9 =B8*D8 =F7-D8 =G7+D8 =(B8-E8)^2 =H8*D8
Total 100 =SUM(D3:D8) =SUM(E3:E8) Variance =SUM(I3:I8)
Average
daily
sales
Total
sales =B3*C3+B4*C4+B5*C5+B6*C6+B7*C7+B8*C8
Total
days =C9
Average =D12/D13
Standard
Deviation =SQRT(I9)
(c) What is the probability that a part selected at random:
1. Was produced by Machine W and should be reworked?

=
2. Was produced by Machine Z and is not satisfactory?
=
3. Was produced by Machine Y and should be scrapped?
=
4. Needs to be reworked?
=
5. Needs to be scrapped given that it was produced by machine W?
=
(d) μ = 4000; σ = 500.
1. What is the probability that sales will be greater than 4250 apples?
, where

=>
2. What is the probability that sales will be less than 3600 apples?
, where
=>
3. What is the probability that sales will be less than 4500 apples?
, where
=>
QUESTION 2 Research Question
1. What is the average age of the Australian population?
The median age in Australia is 37 years (Australian Bureau of Statistics).
2. What is the average age to die in Australia (for both men and women)?
According to the latest WHO data published in 2018, total life expectancy in
Australia is 82.9 years ("Life Expectancy in Australia", 2019).
3. What percentage of people work in Australia?
Employment Rate in Australia was 62.40 percent as at January 2019
(Australian Bureau of Statistics).

QUESTION 3 Statistical Decision Making and Quality
Control
(a) Mean time, μ = 20 hours, standard deviation, σ = 5 hours. Daily labour time ins
normally distributed. Sample size, n = 64
1. At 95% confidence level, α = 0.05
=> => ;
=>
Therefore,
and
2. With n = 9, then => ;
=>
Therefore,
and
(b) Hypothesis testing
The null and alternate hypothesis are:
The critical value is calculated as:
The test statistic is obtained as:

Since the test statistic, Z=1.944 is greater than z-score = 1.645, we reject the null
hypothesis. Therefore, at 95% confidence level, there is sufficient evidence to support
the claim of age discrimination.
References
Newbold, P., Carlson, W., & Thorne, B. (2013). Statistics for Business and
Economics(8th ed.). New York: Pearson.
Employment and Unemployment - Australian Bureau of Statistics. (2019). Retrieved
from https://www.abs.gov.au/Employment-and-Unemployment
Life Expectancy in Australia. (2019). Retrieved from
https://www.worldlifeexpectancy.com/australia-life-expectancy