Question

Compare the two chocolate companies from the average amount of sugar/per serving in their chocolate cake

Test at .02 significance level

Company A: Average amount of sugar= 25 grams;   standard deviation = 3 grams;    n = 13

Company B:    Average amount of sugar= 30 grams;   standard deviation = 10 grams;   n = 16

Steps to be covered:

1. State the hypotheses, and identify the claim
2. Find the critical value(s) – you might want to draw the curve
3. Compute the test (statistic) value
4. Make the decision to reject or not reject the null hypothesis.

Answer 1

T-test for two Means – Unknown Population Standard Deviations

The provided sample means are shown below:

X1bar = 25 X2bar ​=30

Also, the provided sample standard deviations are:

s13 s2​=10

and the sample sizes are

n1​=13 n2​=16.

1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: μ1​ = μ2​ Two chocolate companies do not have significant difference in the average amount of sugar/per serving in their chocolate cake

Ha: μ1​ ≠ μ2​  Two chocolate companies have significant difference in the average amount of sugar/per serving in their chocolate cake.

This corresponds to a two-tailed test, for which a t-test for two population means, with two independent samples, with unknown population standard deviations will be used.

2) Rejection Region

Based on the information provided, the significance level is α=0.02, and the degrees of freedom are df=27.

Hence, it is found that the critical value for this two-tailed test is tc​=2.473, for α=0.02 and df=27.

The rejection region for this two-tailed test is R={t:∣t∣>2.473}. (using t distribution table)

3) Test Statistics

Since it is assumed that the population variances are equal, the t-statistic is computed as follows:

4) Decision about the null hypothesis

Since it is observed that ∣t∣=1.735≤tc​=2.473, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach: The p-value is p=0.0941, and since p=0.0941≥0.02, it is concluded that the null hypothesis is not rejected.

(Here p value is calculated using t distribution, Due to symmetry p= 2*P[t>1.735] )

5) Conclusion

It is concluded that the null hypothesis Ho is not rejected. Therefore, there is not enough evidence to claim that the population mean μ1​ is different than μ2​, at the 0.02 significance level.

Two chocolate companies do not have significant difference in the average amount of sugar/per serving in their chocolate cake

Graphically