Question

Sleep apnea: Sleep apnea is a disorder in which there are pauses in breathing during sleep. People with this condition must w up frequently to breathe. In a sample of 432 people aged 65 and over, 108 of them had sleep apnea.

Construct a 99.5% confidence interval for the proportion of those aged 65 and over who have sleep apnea. Round the answer to three decimal places.

Answer 1

99.5% confidence interval = [0.191 , 0.309]

Explanation:

We are given that in a sample of 432 people aged 65 and over, 108 of them had sleep apnea.

The pivotal quantity for 99.5% confidence interval for the proportion of those aged 65 and over who have sleep apnea is given by;

P.Q. =
`\frac{\hat p - p}{\sqrt{(\hat p(1-\hat p))/(n) } }` ~ N(0,1)

where,
`\hat p` = sample proportion = 108/432 = 0.25

p = population proportion

n = sample size = 432

So, 99.5% confidence interval for the population proportion, p is given by;

P(-2.813 < N(0,1) < 2.813) = 0.995 {At 0.5% level of significance z table gives

critical value of 2.813}

P(-2.813 <
`\frac{\hat p - p}{\sqrt{(\hat p(1-\hat p))/(n) } }` < 2.813) = 0.995

P(-2.813 *
`{\sqrt{(\hat p(1-\hat p))/(n) }` <
`{\hat p - p}` < 2.813 *
`{\sqrt{(\hat p(1-\hat p))/(n) }` ) = 0.995

P(
`\hat p` - 2.813 *
`{\sqrt{(\hat p(1-\hat p))/(n) }` < p <
`\hat p` + 2.813 *
`{\sqrt{(\hat p(1-\hat p))/(n) }` ) = 0.995

99.5% confidence interval for p = [
`\hat p` - 2.813 *
`{\sqrt{(\hat p(1-\hat p))/(n) }` ,
`\hat p` + 2.813 *
`{\sqrt{(\hat p(1-\hat p))/(n) }` ]

= [0.25 - 2.813 *
`{\sqrt{(0.25(1-0.25))/(432) }` , 0.25 + 2.813 *
`{\sqrt{(0.25(1-0.25))/(432) }` ]

= [0.191 , 0.309]

Therefore, 99.5% confidence interval for the proportion of those aged 65 and over who have sleep apnea is [0.191 , 0.309] .

Answer 2

Answer:The required interval is (0.196, 0.030).

Explanation:

Since we have given that

n = 432

x = 108

So,
`\hat{p}=(x)/(n)=(108)/(432)=0.25`

at 99% level of confidence, z = 2.576

So, Margin of error would be

`z* \sqrt{(pq)/(n)}\\\\=2.576* \sqrt{(0.25* 0.75)/(432)}\\\\=2.576* 0.021\\\\=0.0536`

So, margin of error = 0.0536

Now, confidence interval would be

`\hat{p}\pm 0.0536\\\\=(0.25-0.0536,0.25+0.0536)\\\\=(0.1964,0.3036)\\\\\approx (0.196,0.030)`

Hence, the required interval is (0.196, 0.030).