Question

a) Use the normal distribution to find a confidence interval for a proportion p given the relevant sample results. Give the best point estimate for p, the margin of error, and the confidence interval. Assume the results come from a random sample.

A 90% confidence interval for p given that p^=0.39 and n=450.

Round your answer for the best point estimate to two decimal places, and your answers for the margin of error and the confidence interval to three decimal places.

Best point estimate =
Margin of error = ±
The 90% confidence interval is ____________ to _____________

b) Use the normal distribution and the given sample results to complete the test of the given hypotheses. Assume the results come from a random sample and use a 5% significance level.

Test H0 : p=0.2 vs Ha : p≠0.2 using the sample results p^=0.26 with n=999

Round your answer for the test statistic to two decimal places, and your answer for the p-value to three decimal places.

test statistic = ______________________
p-value = _______________________
Conclusion: Reject or Do not reject H0?

Answer 1

Solution :

Given that,

a) n = 450

Point estimate = sample proportion = =0.39

1 - = 1-0.39 = 0.61

Z_/2 = 1.645

Margin of error = E = Z _{/ 2} * (( * (1 - )) / n)

= 1.645 * ((0.39*(0.61) /450 )

= 0.038

A 90% confidence interval for population proportion p is ,

- E < p < + E

0.39-0.038 < p < 0.39+0.038

0.352< p < 0.428

The 90% confidence interval for the population proportion p is : 0.352 , 0.428

b)

This is the two tailed test .

The null and alternative hypothesis is

H₀ : p =0.2

H_a : p 0.2

n = 999

= 0.26

P₀ =0.2

1 - P₀ = 1-0.2 = 0.80

Test statistic = z

= - P₀ / [P_{0 *} (1 - P₀ ) / n]

=0.26 -0.20 / [0.20*(0.80) /999 ]

= 4.74

P(z >4.74 ) = 1 - P(z <4.74 ) = 0

P-value = 0

= 0.05    

0 < 0.05

Reject the null hypothesis .