MTH220: In a Semiconductor Production Line, a Manufacturing Batch Contains 1000 Pieces of Wafers A Wafer Contains 81 Arrays: Statistical Methods and Inference, Singapore

Question 1

In a semiconductor production line, a manufacturing batch contains 1000 pieces of wafers. A wafer contains 81 arrays.

(a) The number of defective arrays per wafer follows a Poisson distribution with the mean of 5, and this number of defective arrays per wafer is counted for a random sample of 125 wafers. Compute the probability that the average number of defective arrays per wafer will be less than 5.5.

(b) Now suppose that on average the proportion of defective wafers in a large manufacture batch is 0.1. What is the smallest random sample of items that must be taken from the batch for the probability to be at least 0.99 that the proportion of defective wafers in the sample will be less than 0.13?

Question 2

A student is working on a quiz containing 15 multiple-choice questions. The probability that the student gets the correct answer for a multiple-choice question is 𝑝 = 0.3. Let 𝑋 denote the total number of correct answers.

(a) Determine approximately the value of probability 𝑃(𝑋 = 4) by using the central limit theorem with the correction for continuity.

(b) Compute the exact value of the probability 𝑃(𝑋 = 4) without applying the
approximation. Compare it with the answer obtained in Question 2(a).

Question 3

A statistician is interested in the height of people living in a city. Supposing that the height follows a normal distribution with unknown mean 𝜇 and the know variance 𝜎 2, determine the minimum size of random sample must be taken so that the confidence interval for 𝜇 with confidence coefficient 0.95 and length less than 0.01𝜎.

Question 4

Suppose that 9 observations are selected at random from the normal distribution with unknown mean 𝜇 and unknown variance 𝜎 2 . For these 9 observations, 𝑋̅ 𝑛 = 22 and ∑ (𝑋𝑖 − 𝑛 𝑖=1 𝑋̅ 𝑛) 2 = 72.

(a) Apply a suitable hypothesis test and perform the following hypotheses at the significance level of 0.05. Comment on the result of the hypothesis test if we should accept or reject the null hypothesis.
𝐻0: 𝜇 ≤ 20
𝐻1: 𝜇 > 20

(b) Construct the confidence interval for 𝜇 with a confidence level of 0.95.

Question 5

(a) Suppose that 3 persons are playing a dart game. Suppose that person A plays 10 times, and the probability that she will hit the target is 0.3 on average; person B plays 15 times, and the probability that she will hit the target is 0.2 on average; person C plays 20 times, and the probability that she will hit the target is 0.1 on average. Compute the probability that the target will be hit at least 12 times.

(b) Suppose that 16 digits are chosen at random with replacement from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. Compute the probability that their average will lie between 4 and 6.

Question 6

A group of students from the Department of Economics are interested in inflation over time. Table Q6 below shows the prices of several items in a supermarket in 2010 and 2020. They were trying to predict 2020 prices from 2010 prices. A linear regression model is in consideration.

(a) Plot scatterplot of 2010 prices and 2020 prices. Determine the least-squares regression coefficients for predicting 2020 prices from 2010 prices. You may use R.

(b) If an additional species sold for 21.4 in 2010, predict the 2020 selling price using the model developed in Question 6(a).

(c) Calculate the linear correlation coefficient between 2010 prices and 2020 prices. Comment if the linear model is a good choice for this data set.

Question 1

You are asked to perform some statistical analysis on a particular brand of television. After doing some research, you realised that the time, , it takes for this brand of television to spoil is uniformly distributed between and inclusive, where is a positive constant. Answer the following questions and show full details of your workings.

(a) Derive the mean and variance of T.

(b) Find an approximation to the distribution of the sample mean of a large random sample (size ) of T.

(c) For a particular sample of size 50, Find a 95% confidence interval for a.

(d) Assume a = 10, compute the smallest sample size required if we wish to be 95% confident that the sample mean is within the true mean.

Question 2

A fair die is tossed 60 times.

(a) Find the distribution of the mean score (average score of the 60 scores).

(b) Use the distribution in Question 2(a) to compute the probability that the total score (sum of the 60 scores) is less than or equal to 200. Show full details of your work and write down any results used.

Question 3

Foldable bikes have gained popularity during the current Covid-19 pandemic lock-down in the country. One factor influencing the choice of a foldable bike is the time taken to completely unfold the bike.

Various regular users of two foldable bike brands were tested and their times (in minutes) to unfold the bike was recorded as shown in Table Q3.

A two-sample t-test is to be carried out to test if there is any difference between the two brands at 5% significance level.

You may use R to help you perform or check parts of your workings, but all details must be shown.
(a) What are the assumptions needed to carry out the test?

(b) State the null and alternative hypotheses.

(c) Compute the pooled sample variance.

(d) Determine the value of the test statistic.

(e) Determine the critical (rejection) region for the test.

(f) Comment and write out a brief conclusion for the test.

Marking Guide:

(a) The observed values for the two brands are independent;
2 populations follow the normal distribution
2 populations have equal variance

Question 4

Clean Well is a new all-purpose cleaner being test-marketed by placing displays in three different locations within a selected supermarket. The number of 500ml bottles sold in each location in a week is reported below:

At 5% significance level, we perform a test to check if there is a difference in the mean number of bottles sold at the three locations. Displaying all relevant workings, answer the following questions (You may use R to help you perform or check parts of your workings, but all details must be shown.):
(a) State the null and alternative hypotheses.

(b) For each location, compute the mean and standard deviation of the number of bottles sold.

(c) Like the sum of squares used for determining the least square regression equation line, compute SS(treatment) and SS(error) using the values from Question 4 (b) above.

(d) Hence develop the ANOVA table.

(e) What is the decision rule?

(f) Comment on your conclusion

Marking Guide:

Question 5

A sample of nine flights from Australia Melbourne International Airport and eight from New Zealand Christchurch International Airport reported the following numbers of no-shows for the flights.

(a) At a 5% significance level, use a Wilcoxon rank-sum test to test if there are more no-shows for the flights originating in Melbourne.

(b) Why would a parametric test like the two-sample t-test be unsuitable in this case?

Marking Guide

(a) Ho: The number of no-shows is the same for Melbourne and Christchurch
H1: The number of no-shows is larger for Melbourne than for Christchurch

Question 6

A marketing manager aims to investigate whether gender can influence the car preference. The manager compared auto cars and manual cars. To generalize the study, the manager decides to use a general format to represent the observed frequencies, which are shown in Table Q6. An appropriate test is carried to perform the analysis.

(a) State the null hypothesis and alternative hypothesis.

(b) Determine the expected frequencies.

Marking Guide:

(a) Null hypothesis gender and car preference are independent.
Alternative hypothesis gender and car preference are dependent.

(b) Table of expected frequencies: