Sample Proportions

Worksheet 1

Note: Your browser cannot display some mathematical notation as text. Because of this, the symbol p* should be equated with the "p-hat"--the sample proportion--discussed in class and the text. Complaints? Contact Netscape and Microsoft.

You may wish to use the interactive normal tables or the interactive binomial probabilities page. (Opens a new window. You may only use one of these two at a time!)


In each problem, begin by answering the following questions.

  1. Identify the response variable being measured. Is it categorical or quantitative?
  2. Identify the population and the parameter of interest.
  3. Identify the sample and the statistic.
  4. The statistic has a distribution; what is the mean of this distribution? Is the statistic an unbiased estimate of the parameter? What is the standard deviation of the statistic?
  5. Does the statistic have approximately the normal distribution?

Exercise I

Suppose 0.60 (=60%) of all voters (there are over 10,000) in Oswego County intend to vote for candidate Brown in the upcoming election. A poll is taken, 100 voters are selected by SRS. Take p* to be the proportion of sampled voters who intend to vote for Brown. What is the probability of the event p* < 0.50. (That is, what is the probability less than half of the sampled voters intend to vote for Brown? Such a result might lead to an incorrect prediction; namely that Brown will lose.)


Exercise II

Hershey claims that 10% (= 0.10) of all M&Ms are blue. Assume that the M&Ms in a 1 pound bag form an SRS from all M&Ms (for the most part a reasonable assumption). The particular bag we examine contains 320 M&Ms. Let p* be the proportion of sampled M&Ms that are blue. What is the probability that p* is greater than 0.12? (That is, what is the probability that the bag contains more than 12% blue M&Ms?)What proportion of all 1 pound bags contain over 12% blue M&Ms?


Exercise III

Assume that 2% = 0.02 of all products that come off a company's assembly line are defective. The company produces thousands each day. An SRS of 120 of these products is drawn; take to be the proportion of defectives among the sampled items.


Exercise IV

Which treat do dogs prefer, Snausages or Meaty Bones? Naturally some dogs prefer Snausages, others prefer Meaty Bones. However, in general one treat is preferred to the other if more than 50% of all dogs prefer that product. Assume for the sake of argument that dogs are evenly split on the treats-then 0.50 of all dogs prefer Snausages. We take an SRS of 500 dogs and let p* be the proportion of those 500 who prefer Snausages. Find the probability p* is between 0.45 and 0.55 (within 0.05 of 0.50). It is reasonable to conclude one treat is preferred by more dogs in the population than is the other treat if our sample yields a result more than 0.05 from 0.50. Why (not)? How would you determine which treat is preferred by a given dog?


Exercise V

In 1980 there were approximately 1000 Portuguese Water Dogs in the U.S. An SRS of 150 was selected; p* was the proportion of sampled dogs having hip defects (a known result of inbreeding, and therefore a common problem with breeds of small numbers). Assume 15% of all dogs had hip defects.


Exercise VI

Assume that 0.26 of all children in urban schools fail to graduate from high school. An SRS of 400 children is taken in the metropolitan Newark, NJ area. Let p* be the proportion of sampled children who don't finish high school. What is the probability p* is less than 0.20?