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What hypothesis decision is made?

It is hypothesized that 50% of the U.S. population cannot name both of the U.S. Senators from the state in which they reside. A sample of 500 individuals is taken and 218 correctly name both of their Senators. What decision is made at the 1% level of significance and what test statistic was used?
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1 Answer

Hello Anne,
First of all, when it comes to statistic, the letters I am using may be different from what your class is using, but the formulas will be the same.
Sine we are dealing with a comparison of proportions between a population and sample, , we need to do a z-test using the formula:
π(pi) = population proportion or probability (.5)
p = sample proportion of probability (218/500=.436)
σ = standard deviation
The formula to find standard deviation is:
σ = √(π(π-1)/n)
  = √(.5(1-.5)/500
  = √(.25/500)
  = .02236
We take this back to our z formula.
z = (p-π)/σ
  = (.436-.5)/.02236
  = (-.064/.02236)
  = -2.862
The level of confidence we are given is α=.01.  We will use a two tailed test since no direction of difference was expected, meaning .005 in either tail.  By looking at your z distribution, you can see that this corresponds to a z score of ±2.58.
Our calculated z of -2.862 is more extreme than the critical z of ±2.58, we reject the null hypothesis.  The conclusion is that the proportion is a value other than 50%.