A major taxi company is concerned that its proportion of late arrivals has substantially increased in the past month. Historical data shows that on the average 18% of the company taxis have arrived late. In a random sample of 1,240 taxis, 310 taxis have arrived late. If we are conducting a hypothesis test of a single proportion to determine if the proportion of late arrivals has increased: What is the value of the calculated test statistic?
Using a formula for a binomial proportion one-sample z-test with your data included, we have:
z = .25 - .18 -->test value (310/1240 = .25) minus population value (.18)
divided by
√[(.18)(.82)/1240] -->.18 is test value, .82 is 1 - .18, and 1240 is the sample size
Finish the calculation for the value of the test statistic.
6.416
To calculate the test statistic, we can use the formula for the Z-test for a single proportion:
Z = (p - P) / √(P * (1 - P) / n)
Where:
- Z is the test statistic
- p is the sample proportion (proportion of taxis arrived late in the sample)
- P is the population proportion (historical proportion of taxis arrived late)
- n is the sample size
Given the information provided:
- Sample size (n) = 1,240
- Number of taxis arrived late (p) = 310
- Historical proportion (P) = 0.18 (18%)
First, calculate the sample proportion (p):
p = number of taxis arrived late / sample size
p = 310 / 1,240
p ≈ 0.25 (25%)
Now, substitute the values into the formula:
Z = (0.25 - 0.18) / √(0.18 * (1 - 0.18) / 1,240)
Simplify the expression inside the square root:
Z = (0.07) / √(0.18 * 0.82 / 1,240)
Z = 0.07 / √(0.1476 / 1,240)
Z = 0.07 / √0.0001190323
Z ≈ 5.92
Therefore, the value of the calculated test statistic (Z) is approximately 5.92.