ST130: Basic Statistics

Assignment 1, Semester 1, 2021

Due Date: Sunday 13th June, 2021, 5pm (Fiji-time) Total Marks: 60 Weight: 15%

Instructions:

1. All questions are compulsory. Show all relevant working.

2. Please complete this assignment on your own.

3. Write your information (name, ID, campus, etc.) on the cover page.

4. You have to use MS WORD and its math editor to type your assignment. Then convert MS WORD to PDF and upload the PDF document. It must be a single PDF document.

5. All students are to upload the assignment via drop box created on Moodle.

6. Plagiarized assignments will be given a mark of 0 (zero) and will be reported for disciplinary action.

Question 1 Start on a new page (4 marks)

The water bill in dollars per month for 25 customers in Fiji is given below. Construct a grouped frequency distribution with 5 classes.

6.5 15 22.5 16 21

20.5 18 19 26.1 19

21 19.7 16 27 19

22 17.5 24.3 23 30

13.5 24.9 17 29 30.5

(4 marks)

Question 2 Start on a new page (8 marks)

A. The lifespan of 12 randomly selected dogs of a particular breed were studied. The frequency distribution is shown here. Find the standard deviation.

Years Frequency

1-3

4-6

7-9

10-12

13-15 2

3

4

2

1

(4 marks)

B. The following data give the marks obtained by 15 students in a 30 point test.

4 9 7 6 24

11 4 13 10 9

0 8 7 10 17

i. Calculate the 45th percentile.

ii. Calculate the percentile rank of 17.

(2 + 2 = 4 marks)

Question 3 Start on a new page (17 marks)

A. A plane owned by Fiji Link ATR72 has three engines—a central engine and an engine on each wing. The plane will crash because it were within the occasion that the central engine fails and one of the two wing engines fails. The probability of disillusionment in the midst of any given flight is 0.004 for the central engine and 0.007 for each of the wing engines. Anticipating that the three engines work independently, what is the probability that the plane will crash in the

midst of a flight? (3 marks)

B. Jackson and Alice work at a firm's office as the boss and secretary, respectively. The probability that Jackson is in the office at any given time during business hours is 0.72, while the probability that Alice is in the office is 0.4. Given that Alice is in the office, the probability of Jackson being there is 0.66. Determine the probability that at any given time during office hours,

i. Both Jackson and Alice are in the office.

ii. Alice is the office given that Jackson is in the office.

iii. At least one of them is in the office.

(2 + 3+ 2 = 7 marks)

C. In a lottery, you have to select a three-digit number such as 123. During the drawing, there are three bins, each containing balls numbered 1 through 9. One ball is drawn from each bin to form the three-digit winning number.

i. You purchase one ticket with one three-digit number. What is the probability that you will win this lottery?

ii. There are many variations of this lottery. The primary variation allows you to win if the three digits in your number are selected in any order as long as they are the same three digits as obtained by the lottery agency. For example, if you pick three digits making the number 123, then you will win if 123, 132, 213, 231, and so forth, are drawn. The variations of the lottery game depend on how many unique digits are in your number. Consider the following two different versions of this game. Find the probability that you will win this lottery in each of these two situations.

a. All three digits are unique (e.g., 123)

b. Exactly one of the digits appears twice (e.g., 122 or 121)

(2+ 2 + 3 = 7 marks)

Question 4 Start on a new page (11 marks)

A. The number of masks sold per day at a retail store is shown in the table below, with the corresponding probabilities. Find the average of the distribution. If the owner of the retail store wants to be sure that he has enough masks for the next 7 days, how many should the owner purchase?

Number of masks sold X 18 19 20 21 22

Probability P(X) 0.1 0.2 0.3 0.3 0.1

(3 marks)

B. The average bus fare of a student to travel to USP daily is $5.31. If the distribution of bus fares is approximately normal with a standard deviation of $0.31, what is the probability that a

randomly selected bus fare is less than $4.50? (4 marks)

C. The average monthly salary of staffs at USP is $4164 in a recent year. If the salaries are normally distributed with a standard deviation of $360, find the probability that the mean salary for a random sample of 20 staffs is less than $3900. (4 marks)

Question 5 Start on a new page (20 marks)

A. A survey found that out of 150 citizens, 108 said they have received the first doze of Covid-19 injection. Find the 98% confidence interval of the population proportion of citizens who have

received the first doze of Covid-19 injection. (4 marks)

B. A bakery shop owner wishes to find the 90% confidence interval of the true mean cost of a large fruit cake. How large should the sample be if he wishes to be accurate to within $0.12? A previous study showed that the standard deviation of the price was $0.25. (3 marks)

C. The average amount of time a person exercises daily is 22.7 minutes in a population. A random sample of 20 people showed an average of 29.8 minutes in time with a standard deviation of 9.8 minute. At ??0.01, can it be concluded that the average differs from the population average?

(6 marks)

D. The average household income for a recent year in Fiji was $30,000. Five years earlier the average household income was $24,500. Assume sample sizes of 34 were used and the population standard deviations of both samples were $5928. At 5% level of significance is there

enough evidence to believe that the average household income has increased? (7 marks)

THE END

Assignment 1, Semester 1, 2021

Due Date: Sunday 13th June, 2021, 5pm (Fiji-time) Total Marks: 60 Weight: 15%

Instructions:

1. All questions are compulsory. Show all relevant working.

2. Please complete this assignment on your own.

3. Write your information (name, ID, campus, etc.) on the cover page.

4. You have to use MS WORD and its math editor to type your assignment. Then convert MS WORD to PDF and upload the PDF document. It must be a single PDF document.

5. All students are to upload the assignment via drop box created on Moodle.

6. Plagiarized assignments will be given a mark of 0 (zero) and will be reported for disciplinary action.

Question 1 Start on a new page (4 marks)

The water bill in dollars per month for 25 customers in Fiji is given below. Construct a grouped frequency distribution with 5 classes.

6.5 15 22.5 16 21

20.5 18 19 26.1 19

21 19.7 16 27 19

22 17.5 24.3 23 30

13.5 24.9 17 29 30.5

(4 marks)

Question 2 Start on a new page (8 marks)

A. The lifespan of 12 randomly selected dogs of a particular breed were studied. The frequency distribution is shown here. Find the standard deviation.

Years Frequency

1-3

4-6

7-9

10-12

13-15 2

3

4

2

1

(4 marks)

B. The following data give the marks obtained by 15 students in a 30 point test.

4 9 7 6 24

11 4 13 10 9

0 8 7 10 17

i. Calculate the 45th percentile.

ii. Calculate the percentile rank of 17.

(2 + 2 = 4 marks)

Question 3 Start on a new page (17 marks)

A. A plane owned by Fiji Link ATR72 has three engines—a central engine and an engine on each wing. The plane will crash because it were within the occasion that the central engine fails and one of the two wing engines fails. The probability of disillusionment in the midst of any given flight is 0.004 for the central engine and 0.007 for each of the wing engines. Anticipating that the three engines work independently, what is the probability that the plane will crash in the

midst of a flight? (3 marks)

B. Jackson and Alice work at a firm's office as the boss and secretary, respectively. The probability that Jackson is in the office at any given time during business hours is 0.72, while the probability that Alice is in the office is 0.4. Given that Alice is in the office, the probability of Jackson being there is 0.66. Determine the probability that at any given time during office hours,

i. Both Jackson and Alice are in the office.

ii. Alice is the office given that Jackson is in the office.

iii. At least one of them is in the office.

(2 + 3+ 2 = 7 marks)

C. In a lottery, you have to select a three-digit number such as 123. During the drawing, there are three bins, each containing balls numbered 1 through 9. One ball is drawn from each bin to form the three-digit winning number.

i. You purchase one ticket with one three-digit number. What is the probability that you will win this lottery?

ii. There are many variations of this lottery. The primary variation allows you to win if the three digits in your number are selected in any order as long as they are the same three digits as obtained by the lottery agency. For example, if you pick three digits making the number 123, then you will win if 123, 132, 213, 231, and so forth, are drawn. The variations of the lottery game depend on how many unique digits are in your number. Consider the following two different versions of this game. Find the probability that you will win this lottery in each of these two situations.

a. All three digits are unique (e.g., 123)

b. Exactly one of the digits appears twice (e.g., 122 or 121)

(2+ 2 + 3 = 7 marks)

Question 4 Start on a new page (11 marks)

A. The number of masks sold per day at a retail store is shown in the table below, with the corresponding probabilities. Find the average of the distribution. If the owner of the retail store wants to be sure that he has enough masks for the next 7 days, how many should the owner purchase?

Number of masks sold X 18 19 20 21 22

Probability P(X) 0.1 0.2 0.3 0.3 0.1

(3 marks)

B. The average bus fare of a student to travel to USP daily is $5.31. If the distribution of bus fares is approximately normal with a standard deviation of $0.31, what is the probability that a

randomly selected bus fare is less than $4.50? (4 marks)

C. The average monthly salary of staffs at USP is $4164 in a recent year. If the salaries are normally distributed with a standard deviation of $360, find the probability that the mean salary for a random sample of 20 staffs is less than $3900. (4 marks)

Question 5 Start on a new page (20 marks)

A. A survey found that out of 150 citizens, 108 said they have received the first doze of Covid-19 injection. Find the 98% confidence interval of the population proportion of citizens who have

received the first doze of Covid-19 injection. (4 marks)

B. A bakery shop owner wishes to find the 90% confidence interval of the true mean cost of a large fruit cake. How large should the sample be if he wishes to be accurate to within $0.12? A previous study showed that the standard deviation of the price was $0.25. (3 marks)

C. The average amount of time a person exercises daily is 22.7 minutes in a population. A random sample of 20 people showed an average of 29.8 minutes in time with a standard deviation of 9.8 minute. At ??0.01, can it be concluded that the average differs from the population average?

(6 marks)

D. The average household income for a recent year in Fiji was $30,000. Five years earlier the average household income was $24,500. Assume sample sizes of 34 were used and the population standard deviations of both samples were $5928. At 5% level of significance is there

enough evidence to believe that the average household income has increased? (7 marks)

THE END

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