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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?

### 1 Answer by Expert Tutors

Anthony B. | Math, Stats, Calculus, APA TutorMath, Stats, Calculus, APA Tutor
4.9 4.9 (384 lesson ratings) (384)
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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:
z=(p-π)/σ
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%.

Anthony