Question

In January 2011, The Marist Poll published a report stating that 66% of adults nationally think licensed drivers should be required to retake their road test once they reach 65 years of age It was also reported that interviews were conducted on 1, 018 American adults, and that the margin of error was 3% using a 95% confidence level.

a. Verify the margin of error reported by The Marist Poll.
b. Based on a 95% confidence interval, docs the poll provide convincing evidence that more than 70% of the population think that licensed drivers should be required to retake their road test once they turn 65?

Answer 1

a

The Margin of error is correct

b

No the polls does not provide convincing evidence that more than 70% of the population think that licensed drivers should be required to retake their road test once they turn 65.

Explanation:

From the question we are told that

The population proportion is
`p = 66`% = 0.66

The sample size is n = 1018

The margin of error is MOE = 3 % = 0.03

The confidence level is C = 95%

Given that the confidence level is 95% , then the level of significance is mathematically evaluated as

`\alpha = 100 - 95`

`\alpha = 5`%

`\alpha = 0.05`

Next we obtain the critical value of
`(\alpha )/(2)` from the standardized normal distribution table, the value is

`Z_{(\alpha )/(2) } = 1.96`

The reason we are obtaining critical values for
`(\alpha )/(2)` instead of
`\alpha` is because
`\alpha` represents the area under the normal curve where the confidence level (
`1-\alpha`) did not cover which include both the left and right tail while
`(\alpha )/(2)` is just considering the area of one tail which what we required to calculate the margin of error

Generally the margin of error is mathematically represented as

`MOE = Z_{(\alpha )/(2) } * \sqrt{(p (1-p ))/(n) }`

substituting values

`MOE = 1.96 * \sqrt{(0.66 (1-66 ))/(1018) }`

`MOE = 0.03`

`MOE = 3`%

The 95% is mathematically represented as

`p - MOE < p < p +MOE`

substituting values

`0.66 -0.03 < p < 0.66 +0.03`

`0.63 < p < 0.69`

Looking at the confidence level interval we see that the population proportion is between

63% and 69%

shown that the population proportion is less than 70%

Which means that the polls does not provide convincing evidence that more than 70% of the population think that licensed drivers should be required to retake their road test once they turn 65.