Question

I Greatly appreciate it! I just can't figure these out ):

1)Suppose that you are testing whether a coin is fair. The hypotheses for this test are

H₀: p = 0.5

and

H₁: p ≠ 0.5.

Which of the following would be a type I error?

		Concluding that the coin is fair when in reality the coin is fair.
		Concluding that the coin is not fair when in reality the coin is not fair.
		Concluding that the coin is fair when in reality the coin is not fair.
		Concluding that the coin is not fair when in reality the coin is fair.

2)A pilot survey reveals that a certain population proportion p is likely close to 0.62. For a more thorough follow-up survey, it is desired for the margin of error to be no more than 0.03 (with 95% confidence). Assuming that the data from the pilot survey are reliable, what sample size is necessary to achieve this?

3)For a particular scenario, we wish to test the hypothesis H₀ : p = 0.44. For a sample of size 40, the sample proportion p̂ is 0.47. Compute the value of the test statistic z_obs.

4)For a test of

H₀ : p = p₀

vs.

H₁ : p < p₀,

the value of the test statistic z _obs is -1.37. What is the p-value of the hypothesis test?

Answer 1

Solution:

1)

Concluding that the coin is not fair when in reality the coin is fair.

Because , type I error is "Rejecting the null hypothesis when it is true "

2)

Given,

E = 0.03

c = 95% = 0.95

p = 0.62

1- p = 1 - 0.62 = 0.38

Now,

= 1 - c = 1 - 0.95 = 0.05

/2 = 0.025

= 1.96 (using z table)

The sample size for estimating the proportion is given by

n =

= (1.96)² * 0.62 * 0.38 / (0.03²)

= 1005.64551111

= 1006 ..(round to the next whole number)

Answer : n = 1006

3)

Observe H₁ : p < p₀,

Left tailed test

p value = P(Z < test statistic) = P(Z < -1.37) = 0.0853

p value is 0.0853