Answer to Question #105330 in Statistics and Probability for Katenda

Question

QUESTIONS 7, 8 AND 9 ARE BASED ON THE FOLLOWING

Bob randomly selected a sample of 100 children with ASD and found that only 70 of them
are in special need schools.

Q 7

What is the 90% confidence interval estimate of the proportion of children with ASD in special need schools?
(1) (0.6102; 0.7898)
(2) (0.6246; 0.7754)
(3) (0.6038; 0.7962)
(4) (0.6678; 0;8122)
(5) None of the above.

Q 8

What is the lower limit for the 95% confidence interval estimate of the proportion of children in
special need schools?
(1) 0:6102
(2) 0:6246
(3) 0:7754
(4) 0:7898
(5) None of the above.

Q 9

This time Bob want to determine whether the true proportion of ASD children in special need schools in the population is 0.75.
Assume a 5% level of significance.

Which of the following statements is incorrect?
(1) H0 : π = 0.75 against H1 : π ≠ 0.75
(2) The critical value is 1.96
(3) The value of the test statistic is 1.09
(4) The p-value is 0.2502
(5) We do not reject H0

Answer 1

Q7

$$CI(proportion)=(\hat{p}-z_{\alpha/2}\cdot\sqrt{{\hat{p}(1-\hat{p}) \over n}}, \hat{p}+z_{\alpha/2}\cdot\sqrt{{\hat{p}(1-\hat{p}) \over n}})$$

Given that $n=100, \hat{p}=70/100=0.7$

90% confidence interval: $z_{\alpha/2}=1.645$

$$CI(proportion)=(0.7-1.645\sqrt{{0.7(1-0.7) \over 100}}, 0.7+1.645\sqrt{{0.7(1-0.7) \over 100}})$$

$$CI(proportion)=(0.6245,0.7755)$$

(2) (0.6246; 0.7754) 

Q8

95% confidence interval: $z_{\alpha/2}=1.96$

$$\hat{p}-z_{\alpha/2}\cdot\sqrt{{\hat{p}(1-\hat{p}) \over n}}=0.7-1.96\cdot\sqrt{{0.7(1-0.7) \over 100}}\approx$$ $$\approx0.6102$$

(1) 0:6102 

Q9

The following null and alternative hypotheses need to be tested:

$H_0:\pi=0.75$

$H_1:\pi\not=0.75$

This corresponds to a two-tailed test, for which a z-test for one population proportion needs to be used.

$z_c=1.96$

The rejection region for this two-tailed test is $R=\{z:|z|>1.96\}$

The z-statistic is computed as follows:

$$z={\bar{p}-\pi \over \sqrt{\pi(1-\pi)/n}}={0.7-0.75\over \sqrt{0.75(1-0.75)/100}}\approx-1.1566$$

Since it is observed that $|z|=1.1566<1.96=z_c,$ it is then concluded that the null hypothesis is not rejected. Therefore, there is not enough evidence to claim that the population proportion $\bar{p}$ is different than $\pi$ at the $\alpha=0.05$ significance level.

The p-value is $p=0.2474,$ and since $p=0.2474>0.05,$ it is concluded that the null hypothesis is not rejected. Therefore, there is not enough evidence to claim that the population proportion $\bar{p}$ is different than $\pi$ at the $\alpha=0.05$ significance level.

Which of the following statements is incorrect? 

(3) The value of the test statistic is 1.09

(4) The p-value is 0.2502