Hypothesis Tests for a Population Proportion

See POPULATION PROPORTION HYPOTHESIS TEST - JUNETEENTH EXAMPLE       

Previously, we looked at the language and the general methods of hypothesis testing.  Here we look at a worked example where we want to make an inference about a population proportion.

REVIEW: 

In MODULE 5, we learned the criteria for a binomial probability distribution and that

  1. THE MEAN OF A BINOMIAL PROBABILITY DISTRIBUTION IS GIVEN BY mu subscript x equals E open parentheses X close parentheses equals n p
  2. ITS STANDARD DEVIATION IS GIVEN BY sigma subscript x equals square root of n p q end root    

Recall 1:  For a fixed p, as the number of trials n in a binomial experiment increases, the probability distribution of the random variable X becomes more nearly symmetrical and bell-shaped (normal).  If n p q greater or equal than 10 the probability distribution will be approximately symmetrical and bell-shaped (normal).  We had to make a CONTINUITY CORRECTION:  to find P( a ≤ X ≤ b) in the BINOMIAL distribution, we needed to use P( a - 0.5 ≤ X ≤ b + 0.5) in the NORMAL distribution.

Recall 2: For a simple random sample of size n with a population proportion p, three things were true of the SAMPLING DISTRIBUTION of p with hat on top:

  1. The mean of the sampling distribution of is p (so p with hat on top is an unbiased estimator of p).
  2. The standard deviation of the sampling distribution of is square root of fraction numerator p q over denominator n end fraction end root.     (where q = 1-p)
  3. The sampling distribution of p with hat on top is approximately normal.

We also noted that the sampled values are supposed to be independent, a condition we will take to mean that n less or equal than 0.05 N .  We will usually assume that this is true.

And we agreed to disregard the continuity correction WHEN WORKING WITH THE SAMPLING DISTRIBUTION if n > 100. 

In MODULE 6 we used this information to construct a CONFIDENCE INTERVAL FOR A POPULATION PROPORTION.  The binomial distribution is completely determined by n and p. Therefore we use the z-distribution, as long as the conditions noted above are met or assumed.     

Here we use these facts to conduct a HYPOTHESIS TEST FOR A POPULATION PROPORTION. 


You might want to use the "hypothesis test template" distributed in class to make sure that you include ALL the parts of the hypothesis test.  Remember, you must give your conclusion in BOTH "statistics-speak" AND plain English.

I hope that you will find at least one of these videos useful:

www.youtube.com/watch?v=R3Z0TpHHQ9w Links to an external site.

www.youtube.com/watch?v=Pkyb05DZdZg Links to an external site.

 

Here's an example that shows how to use the TI calculator:

www.youtube.com/watch?v=KUmvv0o93RA Links to an external site.

In the next one, "JB" compares two kinds of inference:  a confidence interval and a hypothesis test for the same problem

www.youtube.com/watch?v=M7fUzmSbXWI Links to an external site.