Question

(a) Draw a stem-and-leaf plot of the ages with an increment of 5, and comment on the shape (b) Explain why the stem-and-leaf plot in (a) does not provide evidence against the validity of the use of the t-distribution for these data. Explain further why the plot does not provide evidence against the validity of the sign test, nor the Wilcoxon signed rank test either. (c) Conduct a sign test to determine if the age at which scientists do their best work differs, on average, from 40 years. Conduct the test at α = 0.10, and be sure to write a concluding sentence (d) Repeat (c), but use the Wilcoxon signed rank test. (e) Compare the test results in c), andd). Which one or which ones are you more inclined to believe? Explain based on your knowledge regarding the relative power of the three tests.

Answer 1

a)  

A stem-and-leaf plot of the ages with an increment of 5 is shown below,

Stem	Leaf
2	8
3	1 6 8 8 9 9
4	4 5 5 8
5	4

It is a specific table used to showcase the data in the interesting way. Here each digit in the data is split into a stem. It plots data similar to the horizontal bar graph but uses the original data instead of bars. The stem and leaf plot is similar to histogram in displaying the statistical data.

b)

P(t < -2.196, df = 11) = 0.0252

For two-tail test, p-value = 2 * 0.0252 = 0.0504

Since p-value is less than the significance level of 0.10, we reject H0 and conclude that there is significant evidence that mean age at which scientists do their best work differs from 40 years.

For $\alpha$ = 0.05,

Since p-value is greater than the significance level of 0.05, we fail to reject H0 and conclude that there is no significant evidence that mean age at which scientists do their best work differs from 40 years.

c)

Null hypothesis H0: $\mu$ = 40

Alternative hypothesis H1: $\mu$$\ne$ 40

Standard error of mean =
s2 /n
=
52.2652/12
= 2.087

Test statistic, t = ($\bar{y}$ - $\mu$) / Standard error

= (35.4167 - 40) / 2.087

= -2.196

Degree of freedom = n - 1 = 12 - 1 = 11

For $\alpha$ = 0.10 and df = 11, the critical value of t is $\pm$1.80

The critical region to reject H0 is t < -1.80 or t > 1.80

Since t < - 2.196, we reject H0 and conclude that there is significant evidence that mean age at which scientists do their best work differs from 40 years.

Please post the remaining questions in another post.