Question

(1) A genetic experiment with peas resulted in one sample of offspring that consisted of 417 green peas and 159 yellow peas. (a). Construct a 95​% confidence interval to estimate of the percentage of yellow peas.

b. It was expected that​ 25% of the offspring peas would be yellow. Given that the percentage of offspring yellow peas is not​ 25%, do the results contradict​ expectations?

(2)

Refer to the data set of 2020 randomly selected presidents given below. Treat the data as a sample and find the proportion of presidents who were taller than their opponents. Use that result to construct a​ 95% confidence interval estimate of the population percentage. Based on the​ result, does it appear that greater height is an advantage for presidential​ candidates? Why or why​ not?

PRESIDENT	HEIGHT	HEIGHT OPP
T. Roosevelt	178	175
J. Kennedy	183	182
Harrison	173	168
Cleveland	180	180
Lincoln	193	188
Van Buren	168	180
Eisenhower	179	178
Coolidge	178	180
Jefferson	189	170
Carter	177	183
Pierce	178	196
Polk	173	185
J. Q. Adams	171	191
Hoover	182	180
G. W. Bush	183	185
F. Roosevelt	188	182
Johnson	192	180
Nixon	182	180
Buchanan	183	175
Harrison	168	180

Construct a​ 95% confidence interval estimate of the percentage of presidents who were taller than their opponents.

Answer 1

(1) A genetic experiment with peas resulted in one sample of offspring that consisted of 417 green peas and 159 yellow peas. (a). Construct a 95​% confidence interval to estimate of the percentage of yellow peas.

'n' is the total number of peas

n = 579

X: No. of yellow peas

x = 159

= x /n

(1- )% CI for population proportion

    

   =1 - 0.95 = 0.05

Therefore the critical value at

=

= 1.96.......using normal percentage tables with p = 0.025

Margin of error = Critical value * SE

= *

x	159
n	576
Sample	0.2760
Critcal V	1.9600
Lower L.	0.2395	23.95%
Upper L.	0.3125	31.25%

b. It was expected that​ 25% of the offspring peas would be yellow. Given that the percentage of offspring yellow peas is not​ 25%, do the results contradict​ expectations?

The expected value is 25%, the 95% confidence interval includes 25%, so we can say that there is 95% chance of proportion of yellow can be 25%.

So if it is given that percentage of yellow peas is not 25%, results (based on CI) are not contracting the expectation

PRESIDENT	HEIGHT	HEIGHT OPP
T. Roosevelt	178	175	taller
J. Kennedy	183	182	taller
Harrison	173	168	taller
Cleveland	180	180	shorter
Lincoln	193	188	taller
Van Buren	168	180	shorter
Eisenhower	179	178	taller
Coolidge	178	180	shorter
Jefferson	189	170	taller
Carter	177	183	shorter
Pierce	178	196	shorter
Polk	173	185	shorter
J. Q. Adams	171	191	shorter
Hoover	182	180	taller
G. W. Bush	183	185	shorter
F. Roosevelt	188	182	taller
Johnson	192	180	taller
Nixon	182	180	taller
Buchanan	183	175	taller
Harrison	168	180	shorter

No. of taller presidents

x = 11

n = 20

(2)

Refer to the data set of 2020 randomly selected presidents given below. Treat the data as a sample and find the proportion of presidents who were taller than their opponents. Use that result to construct a​ 95% confidence interval estimate of the population percentage. Based on the​ result, does it appear that greater height is an advantage for presidential​ candidates? Why or why​ not?

Construct a​ 95% confidence interval estimate of the percentage of presidents who were taller than their opponents.

Since this is again binomial proportion interval with same confidence level we use the same C.V.

x	11
n	20
Sample p	0.55
Critcal V	1.96
Lower L.	0.3320	33.20%
Upper L.	0.7680	76.80%

Since there is the true proportion of taller presidents have 95% being in the above interval which includes 0.5 as well it does not appear that greater height is an advantage for presidential​ candidates.

If it was the case then majority meaning more than 50% would be taller presidents.