Question

A traffic and safety Officer was curious to see if there was a difference in the average driving speeds when comparing Parkway (group 1) drivers to Turnpike (group 2) drivers. Assuming that both populations follow a normal distribution and are independent, is there significant evidence to suggest that there is a difference? Evaluate this comparison at a 0.05 significance level. Although the Population Standard Deviations are unknown, assume them to be equal. (Use the six steps of hypothesis testing.

Xbar= 84, s₁ = 10, n = 30

_Xbar2=85, s₂ = 12, n = 25

Answer 1

Solution:-

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

Null hypothesis: u₁ = u ₂
Alternative hypothesis: u₁ u ₂

Note that these hypotheses constitute a two-tailed test. The null hypothesis will be rejected if the difference between sample means is too big or if it is too small.

Formulate an analysis plan. For this analysis, the significance level is 0.05. Using sample data, we will conduct a two-sample t-test of the null hypothesis.

Analyze sample data. Using sample data, we compute the standard error (SE), degrees of freedom (DF), and the t statistic test statistic (t).

SE = sqrt[(s₁²/n₁) + (s₂²/n₂)]
SE = 3.01552
DF = 53.0
t = [ (x₁ - x₂) - d ] / SE

t = 0.33

where s₁ is the standard deviation of sample 1, s₂ is the standard deviation of sample 2, n₁ is the size of sample 1, n₂ is the size of sample 2, x₁ is the mean of sample 1, x₂ is the mean of sample 2, d is the hypothesized difference between the population means, and SE is the standard error.

Since we have a two-tailed test, the P-value is the probability that a t statistic having 53 degrees of freedom is more extreme than - 0.33; that is, less than -0.33 or greater than 0.33.

Thus, the P-value = 0.743

Interpret results. Since the P-value (0.743) is greater than the significance level (0.05), we have to accept the null hypothesis.

From the above test we do not have sufficient evidence in the favor of the claim that there is a difference.