Question

Next time you see an elderly man, check out his nose and ears! While most parts of the human body stop growing as we reach adulthood, studies show that noses and ears continue to grow larger throughout our lifetime. In one study1 examining noses, researchers report "Age significantly influenced all analyzed measurements:" including volume, surface area, height, and width of noses. In a test to see whether males, on average, have bigger noses than females, the study indicates that "p<0.01."
Let group 1 be males and group 2 be females.

(a) State the hypotheses. Your answer should be an expression composed of symbols

H0:_ vs Ha:_

(c) Interpret the conclusion in context.

-We have evidence that males have bigger noses than females, on average.
-We do not have evidence that males have bigger noses than females, on average.
-We have evidence that males and females have the same size noses, on average.
-We do not have evidence that males and females have the same size noses, on average.

Answer 1

Answer:

a) Hypothesis:

$H_{0}$: $\mu_{Nose}^{Male} = \mu_{Nose}^{Female}$

$H_{a}$: $\mu_{Nose}^{Male} > \mu_{Nose}^{Female}$

Where $\mu_{Nose}^{Male}, \mu_{Nose}^{Female}$ represent the average nose size of male and female respectively.

In words,

$H_{0}$: Males and females have the same size noses, on average.

$H_{a}$: Males have bigger noses than females, on average.

b) We have evidence that males have bigger noses than females, on average.

Step-by-step explanation:

a) The study wishes to investigate if males have bigger noses than females, on average. By hypothesis test, the null hypothesis is the hypothesis we usually want to reject. That is, we want to prove that the male and female nose sizes are not the same. Hence, the null hypothesis will assume equality. And we state that:

$H_{0}$: Males and females have the same size noses, on average.

$H_{a}$: Males have bigger noses than females, on average.

b) We are told that the p-value < 0.01. In general, if the assume 5% as the level of significance ($\alpha$). The decision is:

If p-value is less than the $\alpha$, the test is significant otherwise, the test is insignificant.

The given p-value is less than 5% => 0.05, therefore, we say the test is significant. and we conclude that, we have evidence that males have bigger noses than females, on average.