Question

A study compared the maximum speed (km/hour) of two types of children's bikes: brand A and brand B.
A sample of four Brand A bikes resulted in a sample mean of 13.9 km/hour with sample standard deviation
of 1.2247 km/hour. A sample of four Brand B bikes resulted in a sample mean of 12.2 km/hour with sample
standard deviation of 1.009951 km/hour. Assume the populations are normal with equal variances; and the two
populations are independent. Let A be the population mean for the maximum speed of Brand A bikes. Let
B be the population mean for the maximum speed of Brand B bikes. Suppose a hypothesis test is conducted
to look for evidence that the two brands of children's bikes have a di erent population mean maximum speed.
1
1. State the null and alternative hypothesis.
2. Determine the value and distribution of test-statistic.
3. Compute the P-value and determine if we reject null hypothesis, with = 0:05
4. There are three type of con dence intervals: two-taild, one tailed with upper bound, and one tailed
with lower bound. Choose and compute the appropriate con dence interval for the di erence in the
population mean maximum speed. Verify if we should reject null hypothesis with the con dence interval
you calculated.

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.

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 = 0.79371
DF = 6
t = [ (x₁ - x₂) - d ] / SE

t = 2.14

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 6 degrees of freedom is more extreme than -2.14; that is, less than -2.14 or greater than 2.14.

Thus, the P-value = 0.032

Interpret results. Since the P-value (0.032) is less than the significance level (0.05), we cannot accept the null hypothesis.

4) Two tailed confidence interval is C.I = (0.2422, 3.642).

95% confidence interval for the mean is C.I = (0.2422, 3.642).

C.I = 1.70 + 2.447 × 0.79371

C.I = 1.70 + 1.9422

C.I = (0.2422, 3.642)