10. A manufacturer wanted to know if more coupons would be redeemed if they were mailed to the
female in the household rather than the male in the household. A coupon book was sent at
random to either the male or female in a random sample of 50 male-female households. A
month later, a coupon book was sent to the other member of the pair. The manufacturer found
that the mean difference (female-male) in the number of coupons redeemed was 1.5 with a
standard deviation of 4.75.
Do these data provide convincing evidence that the mean number of coupons redeemed is
greater when the coupons were addressed to a female? Use a = 0.01.
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Using the t-distribution, it is found that the data does not provide convincing evidence that the mean number is greater when the coupons were addressed to a female.

What are the hypothesis tested?

At the null hypothesis, it is tested if there is no difference, that is, the mean is of 0, hence:

[tex]H_0: \mu = 0[/tex]

At the alternative hypothesis, it is tested if the mean number is greater for females, that is, the mean is greater than 0, hence:

[tex]H_1: \mu > 0[/tex]

What is the test statistic?

The test statistic is given by:

[tex]t = \frac{\overline{x} - \mu}{\frac{s}{\sqrt{n}}}[/tex]

The parameters are:

[tex]\overline{x}[/tex] is the sample mean.

[tex]\mu[/tex] is the value tested at the null hypothesis.

s is the standard deviation of the sample.

n is the sample size.

In this problem, the parameters are given as follows:

[tex]\overline{x} = 1.5, \mu = 0, s = 4.75, n = 50[/tex].

Hence, the test statistic is given by:

[tex]t = \frac{\overline{x} - \mu}{\frac{s}{\sqrt{n}}}[/tex]

[tex]t = \frac{1.5 - 0}{\frac{4.75}{\sqrt{50}}}[/tex]

t = 2.23

What is the conclusion?

Considering a right-tailed test, as we are testing if the mean is greater than a value, with a significance level of 0.01 and 50 - 1 = 49 df, the critical value is given by [tex]t^{\ast} = 2.4[/tex].

Since the test statistic is less than the critical value for the right-tailed test, the data does not provide convincing evidence that the mean number is greater when the coupons were addressed to a female.

More can be learned about the t-distribution at https://brainly.com/question/26454209