Question

In a Gallup survey of 1,012 randomly selected U.S. adults (age 18 and over), 53% said that they were dissatisfied with the quality of education students receive in kindergarten through grade 12. Use the normal distribution to construct and interpret a 90% confidence interval for the proportion of U.S. adults who are dissatisfied with the education students receive in kindergarten through grade 12. The standard error is SE

Answer 1

The standard error is 0.0157.

The 90% confidence interval for the proportion of U.S. adults who are dissatisfied with the education students receive in kindergarten through grade 12 is (0.5042, 0.5558). This means that we are 90% sure that the true proportion of U.S. adults who are dissatisfied with the education students receive in kindergarten through grade 12 is between 0.5042 and 0.5558.

Explanation:

In a sample with a number n of people surveyed with a probability of a success of
`\pi`, and a confidence level of
`1-\alpha`, we have the following confidence interval of proportions.

`\pi \pm zs`

In which

z is the zscore that has a pvalue of
`1 - (\alpha)/(2)`.

`s = \sqrt{(\pi(1-\pi))/(n)}` is the standard error.

In a Gallup survey of 1,012 randomly selected U.S. adults (age 18 and over), 53% said that they were dissatisfied with the quality of education students receive in kindergarten through grade 12.

This means that
`n = 1012, \pi = 0.53`

The standard error is:

`s = \sqrt{(\pi(1-\pi))/(n)} = \sqrt{(0.53*0.47)/(1012)} = 0.0157`

The standard error is 0.0157.

90% confidence level

So
`\alpha = 0.1`, z is the value of Z that has a pvalue of
`1 - (0.1)/(2) = 0.95`, so
`Z = 1.645`.

The lower limit of this interval is:

`\pi - zs = 0.53 - 1.645*0.0157 = 0.5042`

The upper limit of this interval is:

`\pi + zs = 0.53 + 1.645*0.0157 = 0.5558`

The 90% confidence interval for the proportion of U.S. adults who are dissatisfied with the education students receive in kindergarten through grade 12 is (0.5042, 0.5558). This means that we are 90% sure that the true proportion of U.S. adults who are dissatisfied with the education students receive in kindergarten through grade 12 is between 0.5042 and 0.5558.