Question

a random survey of 1000 U.S. adults in 2007, 570 said it is likely that life exists on other planets. In a recent year, in a random survey of 1000 U.S. adults, 530 said it is likely that life exists on other planets.

a) At a 5% level of significance, can you support the claim that the proportion of U.S. adults who believe it is likely that life exists on other planets is less now than in 2007?

Hypotheses:

Test Statistic:

Analyze data (assuming H0 is true):

State a Conclusion:

b) (2pts) Suppose the actual percentage of adults that believe in life on other planets in 2007 was 55% and those that believe in life on other planets now it is 55%. If this is true, then was the correct decision made in part (a)? If not, what type of error was made?

Answer 1

a) As we are testing here whether the proportion of U.S. adults who believe it is likely that life exists on other planets is less now than in 2007, therefore this is a case of a one tailed test for which the set of hypothesis here are given as:

The sample proportions here are computed as:
p₁ = 570 / 1000 = 0.57
p₂ = 530/1000 = 0.53

The pooled proportion here is computed as:
P = (570 + 530) / 1000 = 0.55

Therefore the standard error here is computed as:

The test statistic now is computed here as:

As this is a one tailed test, the p-value here is computed from the standard normal tables,
p = P(Z > 1.8) = 0.0359

As the p-value here is 0.0359 < 0.05 which is the level of significance, therefore the test is significant here and we can reject the null hypothesis here and conclude that we have sufficient evidence that proportion of U.S. adults who believe it is likely that life exists on other planets is less now than in 2007.

b) As we are given here that the true proportions are equal for both the years, that is 0.55. Therefore we made the incorrect decision in part a) .

Rejecting a True null hypothesis is called a type I error. Therefore we made type I error in part a) here.