Question

Suppose that you are testing the hypotheses Upper H 0 ​: pequals 0.43 vs. Upper H Subscript Upper A ​: pgreater than 0.43. A sample of size 150 results in a sample proportion of 0.48 . ​a) Construct a 99 ​% confidence interval for p. ​b) Based on the confidence​ interval, can you reject Upper H 0 at alpha equals0.005 ​? Explain. ​c) What is the difference between the standard error and standard deviation of the sample​ proportion? ​d) Which is used in computing the confidence​ interval?

Answer 1

Following are the answer to the given points.

Explanation:

In point a:

The confidence interval for p is 95%

using formula:

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

`= 0.28 - 1.96 * \sqrt {((0.28 * 0.72))/(350) } < p < 0.28 + 1.96 * \sqrt{(\frac{0.28 * 0.72)} { 350}}\\\\= 0.233 < p < 0.327`

In point b:

Because 0.22 is not within the trust interval, they have enough proof of H0 at level 0.05.

In point c:

For the percentage for samples,

`\text{Standard error} = \sqrt { (P (1 - p))/(n) }` from ratio p

`\text{Standard deviation} = \sqrt{ \frac{\hat{p}( 1 - \hat{p})}{n} }` from sample ratio
`\hat{p}`

In point d:

Standard deviation is used to measure the interval of confidence

`\text{Standard deviation} = \sqrt{ \frac{\hat{p}( 1 - \hat{p})}{n} }`