Question

The following data lists the ages of a random selection of actresses when they won an award in the category of Best​ Actress, along with the ages of actors when they won in the category of Best Actor. The ages are matched according to the year that the awards were presented. Complete parts​ (a) and​ (b) below.

a. Use the sample data with a

0.010.01

significance level to test the claim that for the population of ages of Best Actresses and Best​ Actors, the differences have a mean less than 0​ (indicating that the Best Actresses are generally younger than Best​Actors).In this​ example,

mu Subscript dμd

is the mean value of the differences d for the population of all pairs of​ data, where each individual difference d is defined as the​ actress's age minus the​ actor's age. What are the null and alternative hypotheses for the hypothesis​ test?

Upper H 0H0​:

mu Subscript dμd

equals=

00 ​year(s)

Upper H 1H1​:

mu Subscript dμd

less than<

00 ​year(s)

​(Type integers or decimals. Do not​ round.)

Identify the test statistic.

tequals=negative 3.68−3.68

​(Round to two decimal places as​ needed.)

Identify the​ P-value.

​P-valueequals=0.0020.002

​(Round to three decimal places as​ needed.)

What is the conclusion based on the hypothesis​ test?

Since the​ P-value is

less than or equal to

the significance​ level,

reject

the null hypothesis. There

is

sufficient evidence to support the claim that actresses are generally younger when they won the award than actors.

b. Construct the confidence interval that could be used for the hypothesis test described in part​ (a). What feature of the confidence interval leads to the same conclusion reached in part​ (a)?

The confidence interval is

nothing

​year(s)less than<mu Subscript dμdless than<nothing

​year(s).

​(Round to one decimal place as​ needed.)

What feature of the confidence interval leads to the same conclusion reached in part​ (a)?

Since the confidence interval contains

zero,

only positive numbers,

only negative numbers,

reject

fail to reject

the null hypothesis.

Actress (years)   Actor (years)
30   58
30   41
31   35
28   42
33   31
27   34
25   46
42   38
29   41
31   43

Answer 1

Here

d = actress's age - actor's age

The hypotheses are :

Null hypothesis against alternative hypothesis

Here

sample mean of difference

sample standard deviation of difference

and sample size

The test statistic can be written as

which under H₀ follows a t distribution with n-1 df

We reject H₀ at 0.01 level of significance if P-value < 0.01

Now,

The value of the test statistic

P-value =

Conclusion :

Since the​ P-value is less than or equal to the significance​ level,reject the null hypothesis. There is sufficient evidence to support the claim that actresses are generally younger when they won the award than actors.

b) a 99% confidence interval estimate for mean difference

So, we can write, at 99% confidence level,

Since the confidence interval contains only negative numbers, reject the null hypothesis.