Question

Your startup company’s app-enabled ketchup and mustard dispenser uses a special plastic gear. The strength of the plastic in the gears (measured in foot-pounds) is important if the dispenser is to last through an entire basketball season. Random samples from two different suppliers are gathered and the following descriptive statistics are measured:

i. Sample 1:

1. n=10

2. ???=??????

3. ????=????

ii. Sample 2:

1. n=16

2. ???=??????

3. ????=????

You believe that the strength of both manufacturer’s products is normally distributed, but you have no reason to believe they exhibit the same variance.

a. The supplier from Sample 2 claims that their gears have a higher mean strength than any other manufacturer in the market. Do the data support this claim at ??=??.?????

b. Find a 95% two-sided confidence interval on the difference in mean strength.

Answer 1

Solution:-

a)

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

Null hypothesis: u₁> u₂
Alternative hypothesis: u₁ < u₂

Note that these hypotheses constitute a one-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 = 6.6821
DF = 24.0

t = [ (x₁ - x₂) - d ] / SE

t = - 4.64

where s₁ is the standard deviation of sample 1, s₂ is the standard deviation of sample 2, n₁ is thesize 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 population means, and SE is the standard error.

The observed difference in sample means produced a t statistic of - 4.64.

Therefore, the P-value in this analysis is less than 0.001.

Interpret results. Since the P-value (almost 0) is less than the significance level (0.05), we can reject the null hypothesis.

From the above test we have sufficient evidence in the favor of the claim that Sample 2 claims that their gears have a higher mean strength than any other manufacturer in the market.

b) 95% two-sided confidence interval on the difference in mean strength is C.I = ( - 44.792, - 17.208).

C.I = 290 - 321 + 2.064 × 6.6821

C.I = - 31.0 + 13.792

C.I = ( - 44.792, - 17.208)