Question

About 78% of all female heart transplant patients will survive at least 3 years. Eighty female heart transplant patients are randomly selected. What is the probability that the sample proportion surviving for at least 3 years will be less than 70%?. Assume the sampling distribution of sample proportion is a normal distribution. The mean of the sample proportion is equal to the population and the standard deviation is equal to square root of pq/n. The probability that the sample proportion surviving for at least 3 years will be less than 70% is _ . Round to 4 decimal places

Answer 1

In this case

`p=0.78`

`n=80`
`np=62.4`
`q=1-p=0.22`
`nq=17.6`
`\text{Mean}=0.78=0.78`
`\text{StandarDeviation}=\sigma=\sqrt[]{(0.78\cdot0.22)/(80)}\approx0.04631`

And the probability that the sample proportion surviving for at least 3 years will be less than 70% is approximately 0.0421

The Z-score in this case is:

`Z=(0.78-0.7)/(0.04631)=1.7273`