Question

You are conducting a study to see if the proportion of voters who prefer the Democratic candidate is significantly smaller than 55% at a level of significance of αα = 0.01. According to your sample, 26 out of 56 potential voters prefer the Democratic candidate.

1. For this study, we should use Select an answer z-test for a population proportion t-test for a population mean
2. The null and alternative hypotheses would be:
   Ho: ? μ p  ? ≠ < = >   (please enter a decimal)   
   H₁: ? μ p  ? ≠ > = <   (Please enter a decimal)

3. The test statistic ? t z  =  (please show your answer to 3 decimal places.)
4. The p-value =  (Please show your answer to 4 decimal places.)
5. The p-value is ? ≤ >  αα
6. Based on this, we should Select an answer fail to reject reject accept  the null hypothesis.
7. Thus, the final conclusion is that ...
   • The data suggest the populaton proportion is significantly smaller than 55% at αα = 0.01, so there is sufficient evidence to conclude that the proportion of voters who prefer the Democratic candidate is smaller than 55%
   • The data suggest the population proportion is not significantly smaller than 55% at αα = 0.01, so there is sufficient evidence to conclude that the proportion of voters who prefer the Democratic candidate is equal to 55%.
   • The data suggest the population proportion is not significantly smaller than 55% at αα = 0.01, so there is not sufficient evidence to conclude that the proportion of voters who prefer the Democratic candidate is smaller than 55%.
8. Interpret the p-value in the context of the study.
   • There is a 55% chance of a Type I error.
   • If the population proportion of voters who prefer the Democratic candidate is 55% and if another 56 voters are surveyed then there would be a 9.86% chance fewer than 47% of the 56 voters surveyed prefer the Democratic candidate.
   • There is a 9.86% chance that fewer than 55% of all voters prefer the Democratic candidate.
   • If the sample proportion of voters who prefer the Democratic candidate is 47% and if another 56 voters are surveyed then there would be a 9.86% chance of concluding that fewer than 55% of all voters surveyed prefer the Democratic candidate.
9. Interpret the level of significance in the context of the study.
   • There is a 1% chance that the proportion of voters who prefer the Democratic candidate is smaller than 55%.
   • If the proportion of voters who prefer the Democratic candidate is smaller than 55% and if another 56 voters are surveyed then there would be a 1% chance that we would end up falsely concluding that the proportion of voters who prefer the Democratic candidate is equal to 55%.
   • If the population proportion of voters who prefer the Democratic candidate is 55% and if another 56 voters are surveyed then there would be a 1% chance that we would end up falsely concluding that the proportion of voters who prefer the Democratic candidate is smaller than 55%
   • There is a 1% chance that the earth is flat and we never actually sent a man to the moon.

Answer 1

Given that

n=56

x=26

So

a)

Since we are testing for proportion hence we use Z test for population proportion.

b)

Since we have to test that if proportion is less than 55% or not hence

H0:P=0.55

Ha:P<0.55

c)

Now test statistics is given by

d)

Since test is left tailed test

So

P value =P(Z<-1.303)=0.0960

e)

Since

Alpha =0.01<P value

f)Hence we failed to reject the null hypothesis.

g)

The data suggest the population proportion is not significantly smaller than 55% at αα = 0.01, so there is not sufficient evidence to conclude that the proportion of voters who prefer the Democratic candidate is smaller than 55%.

h)

If the population proportion of voters who prefer the Democratic candidate is 55% and if another 56 voters are surveyed then there would be a 9.86% chance fewer than 47% of the 56 voters surveyed prefer the Democratic candidate

I)

If the population proportion of voters who prefer the Democratic candidate is 55% and if another 56 voters are surveyed then there would be a 1% chance that we would end up falsely concluding that the proportion of voters who prefer the Democratic candidate is smaller than 55%