Question

A random sample of 20 binomial trials resulted in 8 successes. Test the claim that the population proportion of successes does not equal 0.50. Use a level of significance of 0.05.

(a) Can a normal distribution be used for the p hat distribution? Explain.

No, n·p and n·q are both less than 5.

No, n·p is greater than 5, but n·q is less than 5.

No, n·q is greater than 5, but n·p is less than 5.

Yes, n·p and n·q are both greater than 5. Yes, n·p and n·q are both less than 5.

(b) State the hypotheses.

H0: p = 0.5; H1: p > 0.5

H0: p = 0.5; H1: p < 0.5

H0: p = 0.5; H1: p ≠ 0.5

H0: p < 0.5; H1: p = 0.5

(c) Compute p hat. (Enter a number.)

p hat =

Compute the corresponding standardized sample test statistic. (Enter a number. Round your answer to two decimal places.)

___

(d) Find the P-value of the test statistic. (Enter a number. Round your answer to four decimal places.)

_____

(e) Do you reject or fail to reject H0? Explain.

At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.

At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.

At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

(f) What do the results tell you?

The sample p hat value based on 20 trials is not sufficiently different from 0.50 to justify rejecting H0 for α = 0.05.

The sample p hat value based on 20 trials is sufficiently different from 0.50 to not reject H0 for α = 0.05.

The sample p hat value based on 20 trials is sufficiently different from 0.50 to justify rejecting H0 for α = 0.05.

The sample p hat value based on 20 trials is not sufficiently different from 0.50 to not reject H0 for α = 0.05.

Answer 1

given data are sample n =20

x=8 successes

population proportion p =0.50

then q=1-p=0.50

a) np=20×0.50=10

nq=20×0.50=10

we use normal distribution for

yes,np and nq are have both greater than 5

b) H0: p=0.50 vs H1:p0.50

c) =x/n =8/20=0.40

test statistic :- z=

z=(0.4-05)/

=-0.10/0.1118

= -0.8944= - 0.89

d) in 2tail pvalue = 2× p (z <-0.89)

= 2×(0.1867) = 0.3734

e) pcal>p (i.e, 0.3734>0.05) so fail to reject the null hypothesis H0.

At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

f) The sample p hat value based on 20 trials is not sufficiently different from 0.50 to not reject H0 for α = 0.05.