Activity 17

MATH 216: Statistical Thinking

Activity 1: Quiz

  • Question: If the population standard deviation (\(\sigma\)) is 14.43 mm and the sample size (\(n\)) is 11, what is the standard error of the mean (\(\sigma_{\bar{x}}\))?

      1. 1.37 mm
      1. 4.35 mm
      1. 14.43 mm
      1. 43.29 mm

Activity 2: Group Activities with Real Data

  • Task: Using the provided dataset, analyze how the sampling distribution changes as the sample size increases. The population mean and standard deviation are 50 and 10, resp.

  1. Verify the sample means for different sample sizes (e.g., \(n=11\), \(n=30\), \(n=50\)).

  2. Verify the standard errors calculations and comment on the shape of the sampling distributions.

Activity 3: Case Studies and Problem-Solving

Case Study: A manufacturing company produces steel sheets with a target thickness of 175 mm. Due to machine variability, the thickness is uniformly distributed between 150 and 200 mm. The quality control team takes random samples of 36 sheets to monitor the process.

  • Question: What is the probability that the sample mean thickness is greater than 180 mm?

Activity 4: Sampling Distribution

Prompt: Suppose we have selected a random sample of \(n=36\) observations from a population with mean \(\mu=80\) and standard deviation \(\sigma=6\).

  1. What is the value of the mean \(\mu_{\bar{x}}\) and standard deviation (standard error) \(\sigma_{\bar{x}}\) of the sampling distribution of the sample mean \(\bar{x}\)?

  2. What is the probability that the sample mean is greater than 82?

Activity 5: Sampling Distribution of Age

Prompt: Based on data from the U.S. Census, the mean age of college students in 2011 was \(\mu=25\) years, with a standard deviation of \(\sigma=9.5\) years.

  1. Describe the sampling distribution of the mean of a sample of 125 students.

  2. What is the probability that the sample mean age of the 125 students is greater than 26 years?

Activity 6: Sampling Distribution with Non-Normal Populations

Question 1 (Exponential Distribution):

The time between customer arrivals at a coffee shop follows an exponential distribution with a rate parameter \(\lambda = 0.2\) (mean \(\mu = 5\) minutes). If 50 arrival times are randomly sampled:
a. What are the mean (\(\mu_{\bar{x}}\)) and standard error (\(\sigma_{\bar{x}}\)) of the sampling distribution?
b. What is the probability that the sample mean exceeds 6 minutes?

Question 2 (Poisson Distribution):

A call center receives an average of 12 calls per hour (Poisson-distributed). Suppose 100 hourly call counts are sampled:
a. Describe the sampling distribution of the sample mean.
b. Calculate the probability that the sample mean is less than 11.5 calls.