Statistics and Probability Answers

The relationship between age (X, in year) and the number of overseas holidays a year (Y) has been studied using simple linear regression sing the following model.

Yi = 0 + 1 Xi + ui

The following table shows the data of age and the number of overseas destinations for a random sample of 20 respondents.

Age (X)

62

57

40

49

67

54

43

65

54

41

No of overseas holiday (Y)

6

5

4

3

5

5

2

6

3

1

Age (X)

44

48

55

60

59

63

69

40

38

52

No of overseas holiday (Y)

3

2

4

5

4

5

4

2

1

3

The summary of the data is as follows where SST is the total sum of squares, SSR is the regression sum of squares.

Xi = 1060, Yi = 73, Xi2 = 57994, Yi2 = 311, XiYi = 4097,

SST = Σ(𝑌𝑖−𝑌̅)2 = 44.55, SSR = Σ(𝑌̂𝑖−𝑌̅)2 = 28.65

a) Find the estimated regression line 𝑌̂𝑖 = b0 + b1 Xi that links the age with the number of overseas holidays in a year.

b) Test whether coefficient b1 is statistically significant or not. Use 5% significant level.

c) Interpret the estimated slope coefficient, b1.

d) Suppose that Ah Huat is 50 years old, predict how many times he has gone overseas for a holiday?

e) Give one quantitative and one qualitative variable that are expected to affect the number of overseas holiday in a year.

A survey was distributed to a population of 300 targeted participants. Out of 130 respondents, a sample of 106 participants from the automotive industry was selected based on purposive sampling. Additionally, three field interviews were conducted with senior officers in the automotive sector. Survey results were analyzed using Microsoft Excel Statistical Package, while field interviews were analyzed through coding. The findings indicated that applying MBO in the automotive industry delivered better results in terms of productivity and efficiency, especially when supported by investing in technology.

What do you think are the determinants of MBO? (600word limit )

Assume that the mean life of a particular brand of car battery is normally distributed with a mean of 28 months and a standard variation of 4 months.

(a) for a randomly selected battery of this make, what is the probability that it will last between 30 and 34 months?.

(b) what is the probability that a randomly selected battery of this make will fail within years of date of purchase?.

(c) after what time period will 60% of all batteries of this makes fail?.

3.1. For a randomly selected battery of this make, what is the probability that it will last between 30 and 34 months? 

3.1. For a randomly selected battery of this make, what is the probability that it will last between 30 and 34 months? 

a supermarket has been selling discounted apples in bundles of five at their counters. a random sample of 49 bundles weighs 970 grams on average, with a standard deviation of 70 grams. test the hypothesis that =1000 grams against the alternative hypothesis of < 1000 at 0.06 level of significance.

the fraction of defective items in a large lot is ‘p’. to test the null hypothesis h0: p=0.2, one consider the number ‘f’ of defectives in a sample of 8 items and accept the hypothesis if f≤6, and reject the hypothesis otherwise. what is the probability of type-1 error of the test? what is the probability of type-2 error of the test?

A hospital has 100 beds in 50 semi private rooms. Average occupancy over the past several years has been 70%. Determine the number of projected encounters using the 70% occupancy factor. Determine the number of weighted encounters under the assumption that the assigned weight is 4.0.

Question 5

The number of items rejected daily by a manufacturer because of defects for the last 30 days are:

20, 21, 8, 17, 22, 19, 18, 19, 14, 17, 11, 6, 21, 25, 4, 19, 9, 12, 16, 16, 10, 28, 24, 6, 21, 20, 25, 5, 17, 8

a)     Construct a stem-and-leaf display. What can conclude based on the display?        (7 marks)

b)     Calculate the mean, median, and mode of the data.                                                 (6 marks)

c)     Based on the answers in (b), make a conclusion about the skewness of the data.  (3 marks)

A fast-food chain decided to carry out an experiment to assess the influence of advertising expenditure on sales. Different relative changes in advertising expenditure, compared to the previous year, were made in eight regions of the country, and resulting changes in sales levels were observed the accompanying table shows the results.

Increase in advertising expenditure (%)

0

5

15

20

25

30

35

40

Increase in sales (%)

5

10

18

25

35

50

60

65