Question

Question 1
The household expenditure (HK$) of 12 sampled families with 4 members in Hong Kong in September 2019 were as follows:

                  8095     6745     4427     2192     4697     7549
                  4302     4209     3520     7621     2757     6441

1. (a) Find the sample mean and sample standard deviation of the data.

2. (b) Find the inter-quartile range of the data.

3. (c) Find the 15th percentile and 85th percentile of the data.

4. (d) Use the sampled data as reference, what is the probability that the household expenditure of a family with

   4 members in Hong Kong in September 2019 was more than $5000?

5. (e) Use X to denote the household expenditure in a month and Y to denote the household saving in a month.

   Assume Y = 6200 - 0.4X. Find the mean, standard deviation and the variance of variable Y in this sample.

Answer 1

Solution :

Let X : Household Expenditure of families with 4 members

The given sample contains observations from 12 families. Hence n = 12

a. )

Sample Mean :

Sample standard deviation:

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First sort the data in increasing order

2192 2757 3520 4209 4302 4427 4697 6441 6745 7549 7621 8095

Now calculate quartiles and percentiles based on these sorted observations

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b.)

Interquartile Range = Q3 - Q1

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c.)

15 th percentile is given as follows

85 th percentile is given as follows

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d.)

The probability that the household expenditure of a family with

4 members in Hong Kong in September 2019 was more than $5000

= P ( X > 5000 )

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e.) We have Y = 6200 - 0.4 X

Hence there will be 12 observations of Y

Therefore , n= 12

The mean of Y is given as follows :

The standard deviation of Y is calculated as follows :

And the variance of Y is given as follows :

We know that

Therefore the variance of Y is

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Answers :

a.)

Sample Mean = 5212.917

Sample standard deviation = 2003.699

b.)

Inter-Quartile Range = 2909.25

c.)

15 th percentile = 3252.95

85 th percentile = 7574.2

d.)

P(x>5000) = 0.5423

e.)

Mean of Y = 4114.833

Standard deviation of Y = 801.4796

Variance of Y = 642369.6

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