Question

Gather a sample of the proportion of drivers that drive a maroon vehicle. Test the claim that the proportion is greater than 5%. Gather this data by observing and counting the number of vehicles on different roadways and count how many of them are maroon. Try to get a sample size of n=100, or thereabouts. Conduct a hypothesis test for your data at a significance level of 0.05 and construct a 95% confidence interval for the true population proportion.

Answer 1

we observed a total of 100 vehicles and found that 25 of them were maroon in colour.

therefore, the sample size, n = 100

the sample proportion, = 25/100 = 0.25

given: significance level, = 0.05

We will perform the hypotheses test in the following 5 steps:

STEP#1: We must check that the sample is sufficiently large to validly perform the test.

condition that the sample be large is that the following interval lie wholly within the interval [0,1].

[ , ] where is the sample proportion and n is the sample size.

substituting the vales, we get,

[0.25 - 3 , 0.25 + 3]

[0.25 - 3 , 0.25 + 3]

[0.25 - 3 , 0.25 + 3]

[0.25 - 3*0.04, 0.25 + 3*0.04]

[0.13, 0.37]

since, the above interval lies wholly within [0,1], we can say the sample is sufficiently large.

STEP#2: Now the relevant test is:

null hypotheses, H₀: p = p₀ i.e.,p = 0.05 

alternative hypotheses,  H_a: p > p₀ i.e., p > 0.05 @ =0.05

where p denotes the proportion of the maroon vehicles.

STEP#3: Calculating the test statistic:

The test statistic is:

Z =

and has the standard normal distribution.

substituting the values and calculating the test statistic, we get,

Z =

Z =

Z =

Z =

Z = 9.17

STEP#4: Determining the rejection region

Since the symbol in H_a is “>” this is a right-tailed test, so there is a single critical value, z=z_0.05.

From the cumulative normal probability table we read off z=z_0.05 as 1.645. Hence, the rejection region is [1.645,).

STEP#5: Making the decision

as our test statistic, Z = 9.17 is 1.645 < 9.17 < .

we see the test statistic falls in the rejection region.

The decision is to reject H₀. In the context of the problem our conclusion is:

The data provide sufficient evidence, at the 5% level of significance, to conclude that the proportion of drivers driving maroon vehicles is greater than 5%.

95% confidence interval:

CI is calculated by the formula:

CI = [ , ]    = z_0.025 = 1.96

CI = [0.25 - 1.96 , 0.25 + 1.96]

CI = [0.25 -1.96 , 0.25 + 1.96]

CI = [0.25 - 1.96 , 0.25 + 1.96]

CI = [0.25 - 1.96*0.04, 0.25 + 1.96*0.04]

CI = [0.25 - 0.0784, 0.25 + 0.0784]

CI = [0.17, 0.33]