two bus lines/arrival times

Question

A comparison is made between two bus lines to determine if arrival times of their regular buses from Denver to Durango are off schedule by the same amount of time. For 51 randomly selected runs, bus line A was observed to be off schedule an average time of 53 minutes, with standard deviation 19 minutes. For 61 randomly selected runs, bus line B was observed to be off schedule an average of 61 minutes, with standard deviation 17 minutes. Do the data indicate a significant difference in average off-schedule times? Use a 5% level of significance.

What is the value of the sample test statistic? (Test the difference 𝜇₁ − 𝜇₂. Round your answer to three decimal places.)

Estimate the P-value.

P-value > 0.500

0.250 < P-value < 0.500    

0.100 < P-value < 0.250

0.050 < P-value < 0.100

0.010 < P-value < 0.050P-value < 0.010

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

Answer 1

State null and alternate hypotheses

Null Hypothesis (H₀): µ₁-µ₂=0

Alternate Hypothesis (H_a): µ₁-µ₂≠0

Determine z-statistic for two samples

Z = [(x̄₁-x̄₂)-(μ₁-μ₂)]/√[σ₁²/n₁+σ₂²/n₂]

Z = [(53-61)-0]/√[19²/51+17²/61]

Z ≈ -2.33

Calculate p-value by determining area of non-rejection region

P(-2.33<Z<2.33) = P(Z<2.33)-P(Z<-2.33) = 0.9801 - 0.0099 = 0.9802 > 0.500 (P>0.500 is correct).

Therefore, given our significance level of α=0.05, since 0.9802>0.05, we fail to reject the null hypothesis. Hence, we can conclude the data don’t indicate a significant difference in average off-schedule times.

Hope this helped!