Question

3 A researcher Comypanies d and B were each called at 50 randomly selected times The calls to compamy A were independently of the calls to company B. The response times were were as follows was interested in comparing the response times of two different cab companies recorded and the summary statistics Mean response time Standard deviation 7.6 mins 4 mirs 6.9 mins 1.7 mins a Use a 0.02 sigmificance level to test the claim that the mean response time for company A differs p the mean response time for company B b) Construct a 98% confidence interval for egerence at the-

Answer 1

Solution:-

a)

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

Null hypothesis: u₁ = u ₂
Alternative hypothesis: u₁ $\neq$ 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 0.02. 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 = 0.31145
DF = 98
t = [ (x₁ - x₂) - d ] / SE

t = 2.25

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 98 degrees of freedom is more extreme than 2.25; that is, less than -2.25 or greater than 2.25.

Thus, the P-value = 0.027

Interpret results. Since the P-value (0.027) is greater than the significance level (0.02), 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 significance difference between mean response of compnay A and company B.

b) 98% confidence interval for difference in means is C.I = (- 0.037 , 1.437). $C.I = \left (\bar{x}_{1}-\bar{x}_{2} \right )\pm t_{\alpha /2}\times \sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}}$

C.I = 0.70 + 2.365 × 0.31145

C.I = 0.70 + 0.73658

C.I = (- 0.037 , 1.437)