Suppose an article describes results from a representative sample of American adults in 2018 and an
independently selected representative sample of American adults taken in 2019. Survey participants were
asked the following question.
If you could live anywhere you wished, where would you prefer to live—in a big city, small
city, suburb of a big city, suburb of a small city, or in a town or rural area?
In 2018, 393 of the 1,000 people surveyed selected town or rural area. In 2019, 484 of the 1,000 people
surveyed selected town or rural area. Use a significance level of 0.01 to determine if there is convincing
evidence that the proportion of all American adults who prefer to live in a rural area increased from 2018 to
2019. (Let p be the population proportion of American adults in 2018 who prefer to live in a rural area and
p be the population proportion of American adults in 2019 who prefer to live in a rural area.)
a) State the appropriate null and alternative hypotheses.
b) Find the test statistic. (Round your answer to two decimal places.)
c) Use technology to find the P-value.
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a) The appropriate null and alternative hypotheses are:
- Null hypothesis (H0): p1 = p2 (There is no change in the proportion of American adults who prefer to live in a rural area from 2018 to 2019)
- Alternative hypothesis (Ha): p1 < p2 (There is an increase in the proportion of American adults who prefer to live in a rural area from 2018 to 2019)
b) The test statistic can be calculated using the formula:
z = (p1 - p2) / sqrt((p1(1-p1)/n1) + (p2(1-p2)/n2))
where:
p1 = proportion of American adults who prefer to live in a rural area in 2018
p2 = proportion of American adults who prefer to live in a rural area in 2019
n1 = sample size in 2018
n2 = sample size in 2019
In this case:
p1 = 393/1000 = 0.393
p2 = 484/1000 = 0.484
n1 = 1000
n2 = 1000
Plugging in these values, the test statistic is:
z = (0.393 - 0.484) / sqrt((0.393(1-0.393)/1000) + (0.484(1-0.484)/1000))
= (-0.091) / sqrt(0.000238243 + 0.000249276)
= -0.091 / sqrt(0.000487519)
= -0.091 / 0.022086
≈ -4.12
c) To find the P-value, we can use technology or a standard normal distribution table. Using technology, we can find that the P-value is approximately 0.000034, which is much smaller than the significance level of 0.01. Therefore, we have convincing evidence to reject the null hypothesis and conclude that there is an increase in the proportion of American adults who prefer to live in a rural area from 2018 to 2019.
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