Question

Suppose that researchers study a sample of 50 people and find that 10 are left-handed.

(a) Find a 95% confidence interval for the population proportion that is left-handed.

(b) What would the confidence interval be if the researchers used the Wilson value ~p instead?

(c) Suppose that an investigator tests the null hypothesis that the population proportion is 18% against the alternative that it is less than that. If = 0:05 then nd the critical value ^pc. Using ^p as the sample estimate, would the investigator reject the null?

(d) Suppose that researchers are using this critical value but, unbeknownst to them, the true, population proportion is 0.16. Find the power of the test.

Answer 1

a)

p^ +- z* sqrt(p^q^/n)

Population PROPORTION Confidence Interval
	number of success	10
	sample size =	50
	sample proportion =	0.2
	confidence level =	95%
	Number of Decimals in Results =	4
	standard error =	0.0566
	Critical z-value =	1.959964
	Confidence Interval =	0.2	+/-	0.110872305947974
	=	( 0.089127694052026, 0.310872305947974)
	≈	( 0.0891, 0.3109)
	We are 95% confident the true population proportion is between 0.0891 and 0.3109.

b)

c)

Population PROPORTION Hypothesis Test
	X	10
	n = sample size =	50
	phat = sample proportion =	0.2000
	α = alpha =	0.05
Step 1: State Ho and Ha
	Population parameter comparison value?			0.18
	Is this a LEFT, RIGHT or TWO tailed test?			LEFT-tailed
	Null Hypothesis:	Ho: p ≥ 0.18
	Alternate Hypothesis:	Ha: p < 0.18
Step 2: State Decision Criteria; Find Critical Values
	Reject Ho if z < -1.645
	or Reject Ho if p-value < α = 0.05
Step 3: Calculate Test Statistic
	Test statistic = z =	0.368105087
	p-value =	0.643602561
Step 4: Make Decision
	Do not reject Ho
	Do not reject Ho since p-value (0.6436) IS NOT less than α, alpha, = 0.05.
	Also, Do not reject Ho since the Test statistic (0.36811) does not meet the rejection criteria in STEP 2.
Step 5: Summarize Decision
	There is NOT sufficient statistical evidence to reject Ho.

d)

power = reject the null when null is false

= P(p^ < 0.18 - 1.645* sqrt(0.18*0.82/50) | p = 0.16)

= P(p^ < 0.0906 | p = 0.16)

= P(Z < (0.0906 - 0.16)/sqrt(0.16*0.84/50))

= P(Z < -1.3385 )

= 0.0904

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