Question

Many people who commute to work by car to New York City every day, use either the Lincoln Tunnel or the George Washington Bridge. In a random sample of ?1 = 1663 people who commuted to work using the Lincoln Tunnel, out of a population of ?1 = 49,325, ?1 = 825 or ?̂1 = 0.496 said that they carpooled to work. In a random sample of ?2 =1055 people who commuted to working using the George Washington Bridge, out of a population of ?2 =131,693, ?2 = 530 or ?̂2= 0.502 said that they carpooled to work. Conduct the appropriate hypothesis test to determine if there is sufficient evidence to conclude that the proportion of carpoolers commuting to work via the Lincoln Tunnel is less than the proportion of carpoolers commuting to work via the George Washington Bridge. Use α = 0.01
a. Step 1: Verify the assumptions of the Distribution of the Difference between Two Independent Sample Proportions, ?̂1- ?̂2 (3 pts):
• Both samples are randomly selected, or obtained through a randomized experiment
• Both samples are normally distributed, if:
o ?1?̂1(1- ?̂1)≥10:
o ?2?̂2(1- ?̂2)≥10:
• Samples are independent, if:
o ?1≤ 0.05 of N
o ?2≤ 0.05 of N
b. Step 2. Determine the hypotheses (1 pt.):
c. Step 3: Determine the level of significance, α (1pt.):
d. Step 4a: Determine ?̂ and calculate the test statistic (2 pts):
?̂= ?1+?2?1+?2
?0 = ?̂1- ?̂2 √?̂(1−?̂)√1?1+1?2
e. Step 4b. Determine the p-value associated with the test statistic (1 pt.):
f. Step 5: Compare the p-value to the alpha level α for the hypothesis test (1 pt.):
g. Step 6: State the conclusion in a complete sentence (1 pt.):

Answer 1

Solution:-

a)
Both samples are randomly selected, or obtained through a randomized experiment
Both samples are normally distributed, since
o ?1?̂1(1- ?̂1)≥10
o ?2?̂2(1- ?̂2)≥10
Samples are independent, since
o ?1≤ 0.05 of N
o ?2≤ 0.05 of N

b)

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: P₁> P₂
Alternative hypothesis: P₁ < P₂

Note that these hypotheses constitute a one-tailed test.

c)

Formulate an analysis plan. For this analysis, the significance level is 0.01. The test method is a two-proportion z-test.

Analyze sample data. Using sample data, we calculate the pooled sample proportion (p) and the standard error (SE). Using those measures, we compute the z-score test statistic (z).

d)

p = (p₁ * n₁ + p₂ * n₂) / (n₁ + n₂)

p = 0.4985

SE = sqrt{ p * ( 1 - p ) * [ (1/n₁) + (1/n₂) ] }
SE = 0.01968
z = (p₁ - p₂) / SE

z = - 0.32

where p₁ is the sample proportion in sample 1, where p₂ is the sample proportion in sample 2, n₁ is the size of sample 1, and n₂ is the size of sample 2.

e)

Since we have a one-tailed test, the P-value is the probability that the z-score is less than - 0.32.

Thus, the P-value = 0.374

f) Interpret results. Since the P-value (0.374) is greater than the significance level (0.01), we cannot reject the null hypothesis.

g)

From the above test we can conclude that there is not sufficient evidence to conclude that the proportion of carpoolers commuting to work via the Lincoln Tunnel is less than the proportion of carpoolers commuting to work via the George Washington Bridge.