Session Review
Statistics 226
Supplemental Instruction
Iowa State University / Leader: / Luyun
Course: / Stat 226
Instructor: / Anna Peterson
Date: / 3/30/16

Example1: a researcher believes that the mean score of all third graders in a district is lower than the national mean of 32. We wish to know if the researcher’s claim is correct. A random sample of 44 students is collected and the mean score of these students is found to be 28.91 with a standard deviation of 11.

  1. What is the population parameter of interest (describe in words)?
  1. State the null and alternative hypotheses:
  1. What t-statistic is associated with our observed sample mean?
  1. What is the P (T < -1.86)? That is, what is the probability of seeing a test statistic of -1.86 if indeed the population mean of 32 were true? For now, just draw a picture highlighting the probability we wish to find.

Example 2: a manufacturer of a sprinkler system used for fire protection in office buildings claims that the true average system-activation temperature is 130 °F. A random sample of n = 9 systems when tested yielded a sample mean activation temperature of 131.08 °F and a standard deviation of 1.5 °F. We are told the distribution of all activation temperatures is normal. Does out data contradict the manufacturer’s claim?

  1. Define μ in this scenario.
  1. State the null and alternative hypotheses:
  1. What t-statistic is associated with our observed sample mean?
  1. Illustrate with a picture what the probability of interest would be to try and contradict the manufacturer’s claim.

Example 3: a pain reliever currently being used in a hospital is known to bring a relief to patients with a mean time of 3.5 minutes. A new drug administered to a random sample of 50 patients, yields a mean time to relief of 2.8 minutes with standard deviation of 1.14 minutes. Is there sufficient evidence that the new drug lowers the mean time to pain relief?

  1. Define μ in this scenario.
  1. State the null and alternative hypotheses:
  1. What t-statistic is associated with our observed sample mean?
  1. Illustrate with a picture what the probability of interest would be to show if we have enough evidence to show that the new drug indeed lowers the mean time to pain relief.

Confidence interval: scholarships at US private colleges are funded through individual, corporation and foundation contributions. A private college may offer a scholarship to a particular student, but that scholarship will not be confirmed unless that student enrolls at that college. Define a typical scholarship offer as the median of all scholarships offered at a particular situation. You may assume that the distribution of typical scholarship offers at US private colleges in 2012 is normal. A random sample of size eight was drawn from a list containing typical scholarships for every US private college in 2012. This sample yielded the following responses:

14,500 10,250 17,400 12,150 8,500 16,750 9,800 13,910

(a)Calculate a 99% confidence interval for μ. To find values of s and x, you may either calculate them by hand or by JMP. (s=3277.786)

(b)Provide an interpretation of the above interval within the context of the data.

(c)As it turns out, for this particular sample, the sample standard deviation is equal to the population standard deviation. Calculate a 99% confidence interval for the population mean.

(d)How does your interval in part (c) compare with the one you calculated in part (a)? Why does this make sense? Explain the difference so that someone who has not taken statistics can understand it.