Statistics for laboratory scientists

Homework problems for lecture 10

1. Suppose we measure the log₁₀ cytokine response of 15 mice following some treatment, and observe the sample mean xbar=3.2 and sample SD s=1.5.

   1. If x₁, x₂, ..., x₁₅ are iid Normal(mean=2, sd=1.5), What is the chance of observing a sample mean greater than or equal to 3.2?

   2. If x₁, ..., ₁₅ are iid Normal(mean=3, sd=1.5), What is the chance of observing a sample mean greater than or equal to 3.2?

   3. Repeat (a) for the case the population SD is 3.0.

   4. Repeat (b) for the case the population SD is 3.0.

   5. Repeat (a) for the case the population SD is 1.5 but n=3.

   6. Repeat (b) for the case the population SD is 1.5 but n=3.

   7. Repeat (a) for the case the population SD is 1.5 but n=100.

   8. Repeat (b) for the case the population SD is 1.5 but n=100.

2. Suppose we have 100 independent draws from some population distribution whose shape is unknown but where the population mean is 10 and SD is 2.5. Suppose that n=100 is sufficiently large that for the sample mean to have an approximately normal distribution.

   1. What is the chance that the sample mean is within 0.1 units of the population mean?

   2. What is the chance that the sample mean exceeds the population mean by at least 0.25 units?