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.

Actress (years) 31   25   29   31   35   25   25   42   30   32

Actor (years)    56   40   39   34   29   37   52   35   34   44

a. Use the sample data with a 0.01 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, μd 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?

H0​: μd (1) _____ , _____ years

H1​: μd (2) _____ , _____ years

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

(1) >

<

≠

=

(2) <

=

≠

>

Identify the test statistic.

t= _____ ​(Round to two decimal places as​ needed.)

Identify the​ P-value.

​P-value=_____ ​(Round to three decimal places as​ needed.)

What is the conclusion based on the hypothesis​ test?

Since the​ P-value is (3) _____ the significance​ level, (4) _____ the null hypothesis. There (5)_____ sufficient evidence to support the claim that actresses are generally younger when they won the award than actors.

(3) less than or equal to

greater than

(4) reject

fail to reject

(5) is

is not

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 _____ ​year(s)<μd< _____ ​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 (6) _____ (7) _____ the null hypothesis.

(6) zero,

only negative numbers,

only positive numbers,

(7) reject

fail to reject

Answer 1

(a)

H0: 0

H1: 0

From the given data, the following statistics are calculated:

Actress (X)	Actor (Y)	X - Y = d
31	56	- 25
25	40	- 15
29	39	- 10
31	34	- 3
35	29	6
25	37	- 12
25	52	- 27
42	35	7
30	34	- 4
32	44	- 12

From the d values, the following statistics are calculated:

n = 10

= - 95/10 = - 9.5

s_d = 11.4237

SE = s_d/

= 11.4237/= 3.6125

Test statistic is:

t = /SE

= - 9.5/3.6125 = - 2.63

t score = - 2.63

ndf = 10 - 1 = 9

One Tail - Left Side Test

By Technology, p-value = 0.014

Since the P - value is greater than the significance level, fail to reject the null hypothesis. There is not sufficient evidence to support the claim that the acresses are generally younger when they won the award than actors.

(b)

Confidence interval:

= 0.01

ndf = 10 - 1 = 9

From Table, critical values of t = 2.8214

t SE

= (2.8214 X 3.6125)

= - 9.5 10.1923

= ( - 19.6923, 0.6923)

So, we get:

The confidence interval is - 19.7 years < < 0.7 years

Since the confidence interval contains zero fail to reject the null hypothesis.