Question

 

Generally speaking, would you say that most people can be trusted? A random sample of n₁ = 242 people in Chicago ages 18-25 showed that r₁ = 46 said yes. Another random sample of n₂ = 270 people in Chicago ages 35-45 showed that r₂ = 73 said yes. Does this indicate that the population proportion of trusting people in Chicago is higher for the older group? Use α = 0.05.

(a) What is the level of significance?


State the null and alternate hypotheses.

H₀: p₁ = p₂; H₁: p₁ > p₂

H₀: p₁ = p₂; H₁: p₁ < p₂     

H₀: p₁ = p₂; H₁: p₁ ≠ p₂

H₀: p₁ < p₂; H₁: p₁ = p₂

(b) What sampling distribution will you use? What assumptions are you making?

The Student's t. We assume the population distributions are approximately normal.

The standard normal. We assume the population distributions are approximately normal.  

   The Student's t. The number of trials is sufficiently large.

The standard normal. The number of trials is sufficiently large.

What is the value of the sample test statistic? (Test the difference p₁ − p₂. Do not use rounded values. Round your final answer to two decimal places.)


(c) Find (or estimate) the P-value. (Round your answer to four decimal places.)


Sketch the sampling distribution and show the area corresponding to the P-value.

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α?

At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.     At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

(e) Interpret your conclusion in the context of the application.

Fail to reject the null hypothesis, there is sufficient evidence that the proportion of trusting people in Chicago is higher in the older group.

Reject the null hypothesis, there is insufficient evidence that the proportion of trusting people in Chicago is higher in the older group.     

Reject the null hypothesis, there is sufficient evidence that the proportion of trusting people in Chicago is higher in the older group.

Fail to reject the null hypothesis, there is insufficient evidence that the proportion of trusting people in Chicago is higher in the older group.

Answer 1

H₀: p₁ = p₂; H₁: p₁ < p₂  

The standard normal. We assume the population distributions are approximately normal

x1                =	46	x2                =	73
p̂1=x1/n1 =	46/242=0.1901	p̂2=x2/n2 =	73/270=0.2704
n1                       =	242	n2                       =	270
estimated prop. diff =p̂1-p̂2    =0.1901-0.2704=		-0.0803
pooled prop p̂ =(x1+x2)/(n1+n2)=(46+73)/(242+270)=		0.2324
std error Se=√(p̂1*(1-p̂1)*(1/n1+1/n2) =		0.0374
test stat z=(p̂1-p̂2)/Se =(-0.0803/0.0374)=		-2.15

c)

from excel: p value =normaldist(-2.15)=0.0158
here choose the graph with area shaded in the left tail only

since p value <0.05

.At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.

e)

Reject the null hypothesis, there is sufficient evidence that the proportion of trusting people in Chicago is higher in the older group.