HI6007 Statistics for Business Decisions Online Tutoring

Question 1: Post Graduate Units

A: Mean, Median and Mode, Quartile and Percentile

  1. Mean, Median & Mode

Mean = E(x) = ∑X/n = 1284/20 = 65.9 students’ results

Median = arranging the data in ascending order as follows;

“42 53 54 61 61 61 62 63 64 66 67 67 68 69 71 71 76 78 81 83”

In case of the even data, the middle two values are added and divided by two to get the middle value.

{(n+1)/2}th value = 66 + 67 / 2 = 66.5

Mode = the most recurring number (high frequency) is the mode = 61

  1. First & Third Quartile

First Quartile = (n+1) x 1/4 = (20 + 1)/4 = 5.25th = (5th + 6th)/2 = (61 + 61)/2 = 61

Third Quartile = (n+1) x 3/4 = (20 + 1) x (3/4) = 15.75th value = (15th + 16th)/2 = (71+71)/2 = 71

  1. 9oth Percentile

Computing the position L = K/100 x N, whereas N is the total number of observations

L = (90/100) x 20 = 18th value

Hence, the 90th percentile is 78

B: Inferential Statistics

Inferential statistics take the data from a sample and then makes the inference about the larger population from which sample is taken. The most common inferential statistics include confidence intervals, regression analysis and hypothesis testing. For example, one might stand in a mall and ask questions from a sample of 100 people about whether they like to shop at Sephora or not.

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Question 2: Holmes Institute

A: Joint Probability

  1. Table
  Yes No Total
≤ 23 207 201 408
24-26 299 379 678
27-30 185 268 453
31-35 66 193 259
≥ 36 51 169 220
Total 808 1210 2018
  Y  
  Yes No Total
≤ 23 207/2018 = 0.1026 201/2018 = 0.996 408/2018 = 0.2022
24-26 299/2018 = 0.1482 379/2018 = 0.1878 678/2018 = 0.3660
X

27-30

 185/2018 = 0.0917 268/2018 = 0.1328 453/2018 = 0.2245
31-35 66/2018 = 0.0327 193/2018 = 0.0956 259/2018 = 0.1283
≥ 36 51/2018 = 0.0253 169/2018 = 0.0837 220/2018 = 0.1090
Total 808/2018 = 0.4004 1210/2018 = 0.5996 2018/2018 = 1

Every column represents the joint probabilities.

  1. P(24-26/ Yes) = P(24-26 Yes / P Yes = 299/808 or (0.1482/0.4004) = 0.3700495
  2. P(≤ 23/ Yes) = 207/808 = 0.2562

Since, P (≤23/Yes) ≠ P (X≤23). So, events are not independent.

 

 

B: Expected Value and Variance

X P(X) X x P(X) X2 x P(X)
10 0.05 0.5 5
20 0.1 2 40
30 0.1 3 90
40 0.2 8 320
50 0.35 17.5 875
60 0.2 12 720
  P(X) ∑X x P(x) ∑X2 x P(x)
Total Sum 1 43 2050

Mean = ∑X x P(X) = 43

Variance          = E(X2) – [E(X)]2

                        = ∑x2 P (x) – [∑x P (x)]2

                        = 2050 – (43)2

                        = 2050 – 1849

                        = 201

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Question 3: Drugs in Country X

Aim: Carrying out the hypothesis testing

Given: Sample Size = n = 60; Sample Mean =  = 745

Parameter: Let µ be the population mean, so µ is the annual average expenditure per person in the North East Region so  = 838

Let µ be the annual average expenditure from Midwest

Hypothesis:

H0 = the annual average expenditure per person from Midwest doesn’t differ significantly from the expenditure in the Northeast Region

H1 = the annual average expenditure per person from Midwest is lower than in the Northeast region

Symbolically,

H0 = µ =      v/s       H1 = µ

H0 = µ = 838   v/s       H1 = µ  838

Here as population SD is given, so Z-test will be used for testing the hypothesis

Test Statistic:

Z calculated =

So under H0, µ = 838

Z calculated =  = -2.4012

Decision: Reject H0 if Z-cal  Zα

Here 1 – α = 99% = 0.99 = α = 0.01

Z0.01 = 2.32

So here Z-Cal = -2.4012  -2.32

So we reject H0

Conclusion:

The population annual average expenditure per person in Midwest may be lower than in the Northeast region

Answers:

  1. Hypothesis: H0: µ = 838 v/s        H1: µ  838 (one tail test)
  2. We used Z test statistics, because the sample size is more than 30 & population SD is given
  3. Test Statistics = Z-Cal = -2.4012 and P-Value = 0.0082
  4. We reject H0­ if p-value α so p-value = 0.0082  01 so we reject H0
  5. There is a sufficient evidence to conclude that the population annual average expenditure per person in Midwest may be lower than in the Northeast region.

 

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