Question

In Summer 2016, a consulting firm surveyed 100 UMD students for their selections of presidential candidates. Of the 100 students, 21 selected Hilary Clinton, and 16 selected Donald Trump. Based on the sample (assuming that the sample well represents young voters), we can estimate that 1. 95% confidence interval of the percentage of you ng people selecting Hilary is from % to % (answer format: XX.X); 2. 95% confidence interval of the percentage of you ng people selecting Donald Trump is from % (answer format: XX.X); % to id 3. Can you conclude that Hilary will win the election if only young people vote? esr (answer format: "Yes" or "No"). 4. For a political poll of showing who, Hilary Clinton or Donald Trump, will win the election at this moment, we want the margin of errors to be no more than 2 % ( that is, the 95% confidence interval to be a percentage +2%), the suggested minimum sample size will be (answer format: integer XXXX).

Answer 1

Question 1

Confidence interval for Population Proportion is given as below:

Confidence Interval = P ± Z* sqrt(P*(1 – P)/n)

Where, P is the sample proportion, Z is critical value, and n is sample size.

We are given

x = 21

n = 100

P = x/n = 21/100 = 0.21

Confidence level = 95%

Critical Z value = 1.96

(by using z-table)

Confidence Interval = P ± Z* sqrt(P*(1 – P)/n)

Confidence Interval = 0.21± 1.96* sqrt(0.21*(1 – 0.21)/100)

Confidence Interval = 0.21 ± 1.96* 0.0407

Confidence Interval = 0.21± 0.0798

Lower limit = 0.21 - 0.0798 = 0.130

Upper limit = 0.21 + 0.0798 = 0.290

Answer: 13.0% to 29.0%

Question 2

Confidence interval for Population Proportion is given as below:

Confidence Interval = P ± Z* sqrt(P*(1 – P)/n)

Where, P is the sample proportion, Z is critical value, and n is sample size.

We are given

x = 16

n = 100

P = x/n = 16/100 = 0.16

Confidence level = 95%

Critical Z value = 1.96

(by using z-table)

Confidence Interval = P ± Z* sqrt(P*(1 – P)/n)

Confidence Interval = 0.16 ± 1.96* sqrt(0.16*(1 – 0.16)/100)

Confidence Interval = 0.16 ± 1.96* 0.0367

Confidence Interval = 0.16 ± 0.0719

Lower limit = 0.16 - 0.0719 = 0.088

Upper limit = 0.16 + 0.0719 = 0.232

Answer: 8.8% to 23.2%

Question 3

Answer: Yes

Because 95% confidence interval for Hilary exceeds the 95% confidence interval for Donald Trump.

Question 4

Based on previous discussion, Hilary Clinton will win.

So, we have estimate for p = 0.21, q = 1 – 0.21 = 0.79

Margin of error = E = 2% = 0.02

Confidence level = 95%

Critical Z value = 1.96

(by using z-table)

Sample size formula is given as below:

n = p*q*(Z/E)^2

n = 0.21*0.79*(1.96/0.02)^2

n = 1593.304

Required sample size = n = 1594