Answer:
(a) The standard error is 0.0080.
(b) The margin of error is 1.6%.
(c) The 95% confidence interval for the percentage of all young people who earned a high school diploma is (88.4%, 91.6%).
(d) The percentage of young people who earn high school diplomas has increased.
Step-by-step explanation:
Let <em>p</em> = proportion of young people who had earned a high school diploma.
A sample of <em>n</em> = 1400 young people are selected.
The sample proportion of young people who had earned a high school diploma is:
![\hat p=0.90](https://tex.z-dn.net/?f=%5Chat%20p%3D0.90)
(a)
The standard error for the estimate of the percentage of all young people who earned a high school diploma is given by:
![SE_{\hat p}=\sqrt{\frac{\hat p(1-\hat p)}{n}}](https://tex.z-dn.net/?f=SE_%7B%5Chat%20p%7D%3D%5Csqrt%7B%5Cfrac%7B%5Chat%20p%281-%5Chat%20p%29%7D%7Bn%7D%7D)
Compute the standard error value as follows:
![SE_{\hat p}=\sqrt{\frac{\hat p(1-\hat p)}{n}}](https://tex.z-dn.net/?f=SE_%7B%5Chat%20p%7D%3D%5Csqrt%7B%5Cfrac%7B%5Chat%20p%281-%5Chat%20p%29%7D%7Bn%7D%7D)
![=\sqrt{\frac{0.90(1-0.90)}{1400}}\\](https://tex.z-dn.net/?f=%3D%5Csqrt%7B%5Cfrac%7B0.90%281-0.90%29%7D%7B1400%7D%7D%5C%5C)
![=0.008](https://tex.z-dn.net/?f=%3D0.008)
Thus, the standard error for the estimate of the percentage of all young people who earned a high school diploma is 0.0080.
(b)
The margin of error for (1 - <em>α</em>)% confidence interval for population proportion is:
![MOE=z_{\alpha/2}\times SE_{\hat p}](https://tex.z-dn.net/?f=MOE%3Dz_%7B%5Calpha%2F2%7D%5Ctimes%20SE_%7B%5Chat%20p%7D)
Compute the critical value of <em>z</em> for 95% confidence level as follows:
![z_{\alpha/2}=z_{0.05/2}=z_{0.025}=1.96](https://tex.z-dn.net/?f=z_%7B%5Calpha%2F2%7D%3Dz_%7B0.05%2F2%7D%3Dz_%7B0.025%7D%3D1.96)
Compute the margin of error as follows:
![MOE=z_{\alpha/2}\times SE_{\hat p}](https://tex.z-dn.net/?f=MOE%3Dz_%7B%5Calpha%2F2%7D%5Ctimes%20SE_%7B%5Chat%20p%7D)
![=1.96\times 0.0080\\=0.01568\\\approx1.6\%](https://tex.z-dn.net/?f=%3D1.96%5Ctimes%200.0080%5C%5C%3D0.01568%5C%5C%5Capprox1.6%5C%25)
Thus, the margin of error is 1.6%.
(c)
Compute the 95% confidence interval for population proportion as follows:
![CI=\hat p\pm MOE\\=0.90\pm 0.016\\=(0.884, 0.916)\\\approx (88.4\%,\ 91.6\%)](https://tex.z-dn.net/?f=CI%3D%5Chat%20p%5Cpm%20MOE%5C%5C%3D0.90%5Cpm%200.016%5C%5C%3D%280.884%2C%200.916%29%5C%5C%5Capprox%20%2888.4%5C%25%2C%5C%2091.6%5C%25%29)
Thus, the 95% confidence interval for the percentage of all young people who earned a high school diploma is (88.4%, 91.6%).
(d)
To test whether the percentage of young people who earn high school diplomas has increased, the hypothesis is defined as:
<em>H₀</em>: The percentage of young people who earn high school diplomas has not increased, i.e. <em>p</em> = 0.80.
<em>Hₐ</em>: The percentage of young people who earn high school diplomas has not increased, i.e. <em>p</em> > 0.80.
Decision rule:
If the 95% confidence interval for proportions consists the null value, i.e. 0.80, then the null hypothesis will not be rejected and vice-versa.
The 95% confidence interval for the percentage of all young people who earned a high school diploma is (88.4%, 91.6%).
The confidence interval does not consist the null value of <em>p</em>, i.e. 0.80.
Thus, the null hypothesis is rejected.
Hence, it can be concluded that the percentage of young people who earn high school diplomas has increased.