Question

Advertisements for a laundry detergent claim that the colors in clothing washed using this brand remain brighter for longer. To test this claim, a consumer rights group purchased 100 identical brightly colored plaid shirts. Each shirt was randomly assigned to be washed in the advertised detergent or a different brand. After 50 washes, each shirt was compared to a new identical brightly colored plaid shirt. The brightness of each shirt was assigned a score from 0 to 5 by three different judges. The sample mean brightness score for the 58 shirts washed in the advertised detergent was slightly higher than the sample mean score of the 42 shirts washed in other detergents. Assuming the population variances are equal and using the pooled estimate of the standard deviation, a hypothesis test for the difference between the two means would have how many degrees of freedom?

Answer 1

To test the claim of a laundry detergent, a consumer rights group conducted an experiment comparing the brightness of shirts washed in the advertised detergent to a different brand. A hypothesis test for the difference between the means can be conducted using the pooled estimate of the standard deviation, with degrees of freedom equal to the sum of the sample sizes minus two.

Step-by-step explanation:

To test the claim that the colors in clothing washed using the advertised laundry detergent remain brighter for longer, a consumer rights group conducted an experiment. They randomly assigned 100 identical brightly colored plaid shirts to be washed in either the advertised detergent or a different brand. After 50 washes, the brightness of each shirt was scored by three judges. The sample mean brightness score for the 58 shirts washed in the advertised detergent was slightly higher than the sample mean score of the 42 shirts washed in other detergents.

Since the population variances are assumed to be equal, a hypothesis test for the difference between the two means can be conducted using the pooled estimate of the standard deviation. The degrees of freedom for this test can be calculated as (n1 + n2 - 2), where n1 is the sample size for detergent brand 1 and n2 is the sample size for detergent brand 2.

Answer 2

The hypothesis test for the difference between two means, using the pooled estimate of the standard deviation and given the assumption of equal population variances for two independent samples of sizes 58 and 42, would have 98 degrees of freedom.

Step-by-step explanation:

The hypothesis test for the difference between two means with the assumption of equal population variances is an example of a t-test. In this scenario, we are dealing with two independent samples (the shirts washed in the advertised detergent and the shirts washed in other detergents) where the sample sizes are 58 and 42 respectively.

When conducting a two-sample t-test with a pooled estimate of the standard deviation, the degrees of freedom (df) are calculated using the formula:

df = n1 + n2 - 2

Where n1 is the sample size of the first group and n2 is the sample size of the second group. Substituting our sample sizes into the formula, we get:

df = 58 + 42 - 2 = 98

Therefore, the hypothesis test for the difference between the two means would have 98 degrees of freedom.