Reference no: EM133868454

Quantitative Methods

Question 1 - Descriptive Statistics
A customer service manager wants to analyze complaint patterns over the last 12 weeks. The number of complaints received each week was:

3, 4, 6, 2, 4, 5, 6, 7, 3, 5, 4, 6.
They hope to summarize the data to improve operations and staffing.

Question

a) Calculate the mean, median, and mode of the weekly complaints.
b) Compute the range, variance, and standard deviation.
c) Based on your results, discuss whether the distribution appears symmetric or skewed.

Question 2 - Graphical Representation

A statistics lecturer conducted a quiz and compiled student results. The scores of 40 students were grouped as follows:

The lecturer wants to understand performance patterns through visual representation.

Question Final Answer
a) Draw a histogram for the frequency distribution.
b) Construct an ogive and estimate the median score from it.
c) Comment on the shape of the distribution: Is it symmetric, left-skewed, or right-skewed?

Question 3 - Probability with Binomial Distribution

A digital marketing team finds that 30% of promotional emails receive a customer response. To test a new strategy, they send 8 emails and want to analyze expected results.

Question Final Answer
a) Find the probability that exactly 3 emails receive a response.
b) Find the probability that at least 2 emails receive a response.
c) Using a mean of 2.4 and standard deviation of 1.3, evaluate whether outcomes are concentrated or spread out using the empirical rule.

Question 4 - Central Limit Theorem

A utility company monitors monthly household electricity usage, which follows a normal distribution with a mean of 320 kWh and standard deviation of 40 kWh. They want to assess usage patterns from a regional sample.

Question Final Answer
a) What is the probability that a randomly selected household uses more than 350 kWh?
b) If a sample of 25 households is taken, what is the probability the average usage exceeds 330 kWh?
c) Justify your method in part (b) using the Central Limit Theorem.

Question 5 - Hypothesis Testing
A bank claims that loan approvals are processed in 48 hours on average. An internal audit of 36 loans finds a sample mean of 51 hours and standard deviation of 6 hours.

Question Final Answer
a) State the null and alternative hypotheses and identify the appropriate test type.
b) Calculate the test statistic and corresponding p-value at the 5% significance level.
c) Should the bank's claim be rejected based on the data? Explain your reasoning.
Question 6 [6 Marks] - Confidence Intervals and Proportions
In a quality check of 100 light bulbs, 12 were defective. The manager wants to estimate the overall defect rate and check if a later sample supports the same conclusion.

Question Final Answer
a) Construct a 95% confidence interval for the true defect proportion.
b) Interpret the interval in real-world terms. What does it suggest about quality control?
c) If another sample of 100 bulbs has 18 defects, does this support the original result? Why or why not?