Respuesta :
Answer:
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.
Step-by-step explanation:
In a sample with a number n of people surveyed with a probability of a success of [tex]\pi[/tex], and a confidence level of [tex]1-\alpha[/tex], we have the following confidence interval of proportions.
[tex]\pi \pm zs[/tex]
In which
z is the zscore that has a pvalue of [tex]1 - \frac{\alpha}{2}[/tex].
[tex]s = \sqrt{\frac{\pi(1-\pi)}{n}}[/tex] 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 [tex]n = 1012, \pi = 0.53[/tex]
The standard error is:
[tex]s = \sqrt{\frac{\pi(1-\pi)}{n}} = \sqrt{\frac{0.53*0.47}{1012}} = 0.0157[/tex]
The standard error is 0.0157.
90% confidence level
So [tex]\alpha = 0.1[/tex], z is the value of Z that has a pvalue of [tex]1 - \frac{0.1}{2} = 0.95[/tex], so [tex]Z = 1.645[/tex].
The lower limit of this interval is:
[tex]\pi - zs = 0.53 - 1.645*0.0157 = 0.5042[/tex]
The upper limit of this interval is:
[tex]\pi + zs = 0.53 + 1.645*0.0157 = 0.5558[/tex]
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.
