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MS1S460 Mathematical Tools for Computation

Question:

You are the manager of a computer games company. You need to write a report about the following, answering the questions given on the next pages:

1. Analysing sales and customer data.

2. Predicting future performance.

3. Mathematics behind gaming techniques.

This coursework is worth 60% of your module and is marked out of 100. You should attempt all questions. Please make sure to include the cover sheet as the front pages of your report. Your report must be typed. Handwritten scanned work will not be accepted. You must show all workings, giving answers correct to 2 decimal places where appropriate.

(a) (i) Data shows that the number of new loyalty card members follows an arithmetic progression, with each year the number of new members increasing by a common difference. There were 6 new members in the 1st year, and α in the 2nd year.

• How many new members were there in the 10th year?

• What is the total number of members in the 10th year?

• Predict how many years it will take to exceed 7000 members.

(ii) Yearly online gaming sales (in hundreds of pounds) follows a geometric progression, with 3 in the 1st year and α in the 4th year.

• Predict the online gaming sales in the 7th year.

• Predict the total number of online sales for the first 7 years.

(b) The amount spent on our products (in pounds) during the year by a sample of 40 customers is given below:

• Present this data in a frequency table using six equally sized classes.

• Represent this data graphically using a histogram.

• Find the mean and the sample standard deviation of the grouped data.

• Last year’s mean was 52, and the standard deviation was 26.5. Compare this year’s statistics with last year’s, and explain what this means in terms of customer spending.

(c) The data below shows the amount spent in advertising (in thousands of pounds) and the number of units sold (in hundreds) for 7 years.

• Calculate the correlation coefficient and the regression line.

• Produce a scatter plot of the data including the regression line.

• Predict the number of units sold if the company had spent 8 thousand pounds on advertising.

Do you think this prediction is accurate? Explain why.

(i) Our 3 latest games, denoted A, B, and C, have been nominated for separate prizes at an awards ceremony. We estimate that the probability of A, B, or C winning a prize to be 0.5, 0.25 and α/100 respectively. Calculate the probability of:

• All 3 games winning a prize.

• None of our games winning a prize.

• Exactly one of our games winning a prize.

• At least one of our games winning a prize.

(ii) At a separate awards ceremony, our latest game is nominated in 5 categories. The probability of winning one category is α/100. Calculate the probability of:

• Winning in exactly 2 categories.

• Winning in at least 2 categories.

• Winning in all categories.

(b) On average, there are β errors in 500, 000 lines of code, where β is the floor function of α/6 (i.e. divide α by 6 and round down). The latest game in development has 500, 000 lines of code.

• Find the probability of β errors in the code of the latest game. It is thought that the next game in development will have 1, 000, 000 lines of code.

• Find the probability of 2β errors in the code of this game