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 0.05 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 (<, >, =, ≠) _ year(s) H1: μd (<, >, =, ≠) _ year(s) (Type integers or decimals. Do not round) 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 __ (greater than or less than or equal to) the significant level, (reject or fail to reject) the null hypothesis. There _ (is or is not) sufficent 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 thr hypothesis test described in part (A). What feature of the confiudence interval leads to the same conclusion reached in part (a)? The confidence interval is year(s) < μd < year(s) (Round to one deicmal place as needed.) What feature of the confidence interval leads to the same conclusion reached in part (a)? Since the confidence interval contains (only positive numbers or zero or negativ

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Description: 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 th

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Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter8: Sequences, Series, And Probability

Section8.CT: Chapter Test

Problem 24CT: Show the sample space of the experiment: toss a fair coin three times.

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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 0.05 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 (<, >, =, ≠) ___ year(s)

H1: μd (<, >, =, ≠) ___ year(s)

(Type integers or decimals. Do not round)

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 ____ (greater than or less than or equal to) the significant level, ____ (reject or fail to reject) the null hypothesis. There ___ (is or is not) sufficent 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 thr hypothesis test described in part (A). What feature of the confiudence interval leads to the same conclusion reached in part (a)?

The confidence interval is ____ year(s) < μd < ____ year(s)

(Round to one deicmal place as needed.)

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

Since the confidence interval contains ____ (only positive numbers or zero or negative numbers) ____ (reject or fail to reject)

pres
Actress (years) 28
26
30
29
38
25
25 39
30
33
Actor (years)
60
37
32
41
33
31
47
38
34
46
Ise the sample date with o 0 06 gian

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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Given Information:

The 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.

Significance level $α = 0.05$

Claim: 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).

$μ_{d}$ 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.

State the hypothesis as follows:

Null Hypothesis $H_{0} : μ_{d} = 0$

Alternative Hypothesis $H_{1} : μ_{d} < 0$

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