Question

An ecologist finds 220 yellow-flowered plants and 180 white-flowered plants. Use the normal distribution to find the Lower boundary of a 95% confidence interval for the proportion of yellow-flowered plants. Which of the following answers is correct to 2 decimal places?

a. Lower boundary = 0.30
b. Lower boundary = 0.60
c. Lower boundary = 0.50
d. Lower boundary = 0.40

Answer 1

c. Lower boundary = 0.50

Explanation:

In a sample with a number n of people surveyed with a probability of a success of
`\pi`, and a confidence level of
`1-\alpha`, we have the following confidence interval of proportions.

`\pi \pm z\sqrt{(\pi(1-\pi))/(n)}`

In which

z is the z-score that has a p-value of
`1 - (\alpha)/(2)`.

An ecologist finds 220 yellow-flowered plants and 180 white-flowered plants.

220 out of 220 + 180 = 400. So

`n = 400, \pi = (220)/(400) = 0.55`

95% confidence level

So
`\alpha = 0.05`, z is the value of Z that has a p-value of
`1 - (0.05)/(2) = 0.975`, so
`Z = 1.96`.

The lower limit of this interval is:

`\pi - z\sqrt{(\pi(1-\pi))/(n)} = 0.55 - 1.96\sqrt{(0.55*0.45)/(400)} = 0.5`

Thus the correct answer is given by option c.