Question

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In a random sample of 519 judges, it was found that 285 were introverts.

a) Let p represent the proportion of all judges who are introverts. Find a point estimate for p.

b) Find a 99% confidence interval for p. Give a brief interpretation of the meaning of the confidence interval you have found.

c) Do you think the confidence np > 5 and nq > 5 are satisfied in this problem? Explain why this would be an important consideration.

Answer 1

Solution :

Given that,

n = 519

x = 285

Point estimate = sample proportion = = x / n = 285/519=0.549

1 - = 1 - 0.549=0.451

At 99% confidence level the z is ,

  = 1 - 99% = 1 - 0.99 = 0.01

/ 2 = 0.01 / 2 = 0.005

Z_/2 = Z_0.005 = 2.576

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

= 2.576 (((0.549*0.451) /519 )

= 0.0563

A 99% confidence interval for population proportion p is ,

- E < p < + E

0.549- 0.0563< p < 0.549+ 0.0563

0.4927< p < 0.6053

The 99% confidence interval for the population proportion p is : 0.4927, 0.6053