Eroxl's Notes
Written Assignment 2 (STAT 251)

Question 1

A factory produces 50 independent units every hour. Each unit has a 1% probability of being defective. Let the number of defective units in a given hour be .

An inspector examines the production line, and each defective unit has a 3% chance of being detected, independently of others. Let be the number of defective units that are actually detected by the inspector.

(a). Find Mean and Variance of

(b). What is the Probability that there Were at Least 2 Defective Units? (Please round to 3 decimals)

(c). Derive the Distribution of the Number of Detected Defects . Can This Distribution Be Approximated by a Familiar One? If So, Justify Your Reason and State the Parameters. What about ?

Because is very small (almost 0) we can approximate using a Poisson distribution with as the term for such a small . The same reasoning can be used to estimate using another Poisson distribution with .

(d). Given that at Most One Defect Was Detected, what is the Approximate Probability that the True Number of Defective Units . (Please Keep in 3 decimals)

Problem 2

A call centre monitors the number of incoming calls per hour and the time taken by an agent to handle a call. A sample of calls is randomly drawn. The number of calls per hour follows a Poisson distribution with mean . The time to handle a call follows a distribution with mean minutes and standard deviation minutes.

(a). A Random Sample of 36 Calls is Selected. Using the Central Limit Theorem, Find the Probability that the Handling time of the Sample Mean Exceeds 6.5 Minutes

(b). Calculate the Approximate Probability that the Call Center Receives Fewer than 15 Calls in a Randomly Chosen Hour

(c). Suppose 60 Customers Each Independently Have a 10% Chance of Filing a Complaint Calculate the Approximate Probability that More than 8 Complaints Are Received

Problem 3

A statistics professor would like to test whether there is any difference in academic performance between students who are from the faculty of Science and the students from other faculties in his elementary Statistics class. The professor randomly samples 15 students from the faculty of Science and 15 students from other faculties from his class, and has these students write a test.

(a). State the Null and Alternative Hypotheses. Be Sure to Define Your Notation

where is the null hypothesis, is the alternative hypothesis, is the mean of the faculty of science scores, and where is the mean of the other faculties scores.

(b). The Average Exam Score for the Students from Faculty of Science is 80.7, with a Sample Standard Deviation of 9.7, and the Average Exam Score for the Students from other Faculties is 75.1, with a Standard Deviation of 7.6. State the Test Statistic and Its Distribution. Be Sure to Define Your Notation. Compute the Value of the Test Statistic, given the Data Gathered. From the past Data, Professor Assumes that Grades Are Distributed Normally with Equal Population variance for the Two Groups

Where

  • is the sample mean of students in the faculty of science

  • is the sample standard deviation of students in the faculty of science

  • is the sample size of students in the faculty of science

  • is the sample mean of students in the other faculties

  • is the sample standard deviation of students in other faculties

  • is the sample size of students in the other faculties

  • is the test statistic

(c). In order to Be Conservative, the Professor Sets the Significance Level at . Using the Critical Value Approach, what is the professor’s Conclusion for This Test?

Given the significance level we will reject the sample statistic if it's less than or greater than .

Because our test statistic which does not fall within the rejection region we fail to reject the null hypothesis. Hence there is insufficient evidence to conclude that there is a difference in academic performance between students from the faculty of Science and students from other faculties."

Problem 4

Mechanics agree that the chain on a bicycle should be replaced after covering an average of 3000 miles. StakRide is a new bicycle manufacturing company and it wants to check if the chains it produces for bicycles have a longer life span than the existing average. The lead mechanic randomly selects 25 of the new bicycles and test runs them. The resulting sample mean and standard deviation are 3050 miles and 85 miles, respectively.

(a). What Hypothesis Should Be Tested to Determine whether the Life Span of StakRide Bicycle Chains is Longer than the Known Average?

(b). Assuming that the Life Span of the Bicycle Chains is Approximately Normal, what Test Statistic Would You Use to Test the Hypotheses in part(a)? What is the Value of the Test Statistic for This Data?

We use a single sample t-statistic for our test statistic as the population variance is unknown and our sample size is not very large.

(c). What Conclusion Would You Reach for a Significance Level of 0.05

Because our test statistic is above the critical point it falls within the rejection region and thus we reject the null hypothesis in favour of the alternative that "the Life Span of StakRide Bicycle Chains is Longer than the Known Average".