Section A

Calculation Questions

Question 1

A random sample of 500 inmates from the correctional system was selected to be interviewed. They were asked about the monthly bribes they must pay to the guards to receive humane treatments in the prison. The results are summarised in the table below:

1.1 What is the variable of interest in this scenario?
1.2 What are the data type and measurements of scale for the variable of interest from Question 1.1?
1.3 Use the data above to calculate the following statistics:
Note: Round off your answer to 2 decimal places.
a. The mean
b. The fiftieth percentile
c. The mode
d. The standard deviation
e. The Pearson’s skewness and comment on the value
f. The coefficient of variation

1.4 What percentage of the inmates paid between $30 or less than $60 in bribes to the guards from the sample selected?

Question 2

The following table shows 200 employees of a construction company cross-classified based on their position and the types of property that they are currently busy with:

2.1 An employee is selected at random from the company. Calculate the probability that the employee is:
a. A plumber?
b. An electrician and works in commercial property?
c. A builder or works in residential property?
d. An electrician or a plumber?
e. Working in industrial property and a plumber?
f. Working in industrial property, given that the employee is a plumber?
g. Working in both residential and industrial property?

2.2 Answer the following questions:

a. What is the probability of it snowing in December in the Durban CBD?
b. If P(A)=0.2, P(B)=0.5, and P(C)=0.1, what is the probability of P(A ∩ B∩ C), if all three events A, B, and C are independent events?
c. An insurance company believes that the probability of flooding on top of Table Mountian is 0.0001. What is the probability of no flooding on top of Table Mountain?
d. In a deck of 52 cards, half of them are black and the remainder are red, and there are
12 picture cards and the remainder are numbers cards. What is the probability of selecting a red or black card when randomly picking a card from the deck?
e. If an additive gambler plays a national lottery with 36 numbers to select:

i. What is the chance of winning the jackpot (matching 5 numbers)?
ii. What is the chance of selecting non-jackpot winning numbers?
Note: Give your answer infractions.

Question 3

3.1 Canning Town is a blue-collar borough in London, where the borough council tries to squeeze every penny out of the residents. The council set up cameras to observe the parking images in the streets. The records show that 15% of vehicle owners park without pay.

a. What is the likelihood that 3 or 4 vehicles are parked without pay if a random sample of 10 parking images is selected?

b. Calculate the probability that no less than 2 vehicles are parked without pay if 20 parking images are randomly selected.

c. What is the mean (average) number of vehicles that are parked without pay, when one hundred parking images are randomly selected?

3.2 The N3 highway is one of the important links between Durban and Johannesburg. A road maintenance company found that there are, on average, 8 potholes per every 10km.
a. What is the probability that there are 48 potholes per given 60km?
b. What is the likelihood that at most there are 2 potholes per 5km?
c. What is the standard deviation of the number of potholes in the total length of N3 that is around 570km?

Question 4

4.1 The city manager wants to establish the actual mean of units of water used by its residents.

A random sample of 3 000 households was selected and it was found that the average unit of water used was 8.6 units. Assume that the population standard deviation is 2.4 units and the number of units per resident used is normally distributed.

Estimate, with 99% confidence, the actual mean units per resident in the city.

Note:
Round off your answer to two decimal places
Critical value at 90%: 1.645

Critical value at 95%: 1.96
Critical value at 99%: 2.576

4.2 A travel agency wants to find out the actual proportion of young adults who have visited the Kruger National Park in the past two years. A random sample of 3 000 young adults was chosen and it was found that 600 had visited the Kruger National Park in the past two years.

Estimate, with 90% confidence, the actual proportion of young adults who had visited the Kruger National Park in the past two years.

Note:
Round off your answer to three decimal places
Critical value at 90%: 1.645
Critical value at 95%: 1.96
Critical value at 99%: 2.576

4.3 A stokvel wants to establish the actual average loan amount and selects a random sample of 25 of their members. The sample mean was found to be R122, with a sample standard deviation of R12. Assume the loan amount values are normally distributed.

Estimate, with 90% confidence, the actual mean value of the loan amount by all the members.
Note:
Round off your answer to two decimal places
Critical value at 90%: 1.711
Critical value at 95%: 2.064
Critical value at 99%: 2.797

Question 5

5.1 Coding Monkey is a name given to the poorly paid and hardworking programmer in the computer science sector. The main task of the coding monkey involves solving the query assigned to them. The manager claims that it takes less than 1 hour, on average, to solve a query. To test the claim, a sample of 52 queries was randomly selected and it was found that the average time to solve a query is 0.8 hours with a population variance of 0.04 hours.

Use the hypothesis test method to test the claim with a 5% level of significance. (10 Marks)
Critical value at 1%: 2.327
Critical value at 5%: 1.645
Critical value at 10%: 1.282
Round off your answer to 3 decimal places

5.2 A researcher argues that more than 50% of the population support the digital Rand to be issued in South Africa. To test this belief, the researcher sent out 300 randomly selected questionnaires and found that 165 questionnaires stated that they are in support of the digital Rand and the remainder against the idea. Construct a 1% level of significance of the claim that more than 50% of the population support the digital Rand.

Note:
Round off answer to 2 decimal places
10% level of significance: critical value: 1.645
5% level of significance: critical value: 1.96
1% level of significance: critical value: 2.58