Question

a population has a proportion equal to 0.30. calculate the following probabilities with n = 100. find p(overbar(p) ≤ 0.35).

Answer 1

The probability of observing a sample proportion less than or equal to 0.35, given a population proportion of 0.30 and a sample size of 100, is approximately 0.8621 or 86.21%.

To calculate the probability
`P(\bar{p} \leq 0.35)`, where
`\bar{p}` represents the sample proportion, we can use the Central Limit Theorem (CLT) since n = 100 is sufficiently large.

First, we calculate the standard error of the sample proportion, which is given by the formula:
`S E=\sqrt{((p *(1-p)))/(n)}`, where p is the population proportion.

In this case, p = 0.30 and n = 100.

Thus,
`S E=\sqrt{((0.30 *(1-0.30)))/(100)} \approx 0.0458`.

Next, we standardize the sample proportion using the z-score formula:
`z = \frac{(\bar{p} - p)}{SE}`, where
`\bar{p}` is the sample proportion.

In this case, we want to find the probability when
`\bar{p} \leq 0.35`.

So,
`z=((0.35-0.30))/(0.0458) \approx 1.092`.

Using the standard normal distribution table, we find that the probability
`P(Z\leq 1.092)` is approximately 0.8621.

Therefore, the probability
`P(\bar{p} \leq 0.35)` is approximately 0.8621 or 86.21%.

Answer 2

1. P(X ≤ 0.35):

For a normal distribution, we can calculate the z-score of X:

z = (X - μ) / σ = (0.35 - 0.30) / sqrt(0.3 * 0.7 / 100) ≈ 0.71

Using a standard normal distribution table or calculator, we find the area to the left of z = 0.71:

P(X ≤ 0.35) ≈ 0.7611

2. P(X > 0.40):

Calculate the z-score of :X

z = (0.40 - 0.30) / sqrt(0.3 * 0.7 / 100) ≈ 1.41

Using a standard normal distribution table or calculator, we find the area to the right of z = 1.41:

P(X > 0.40) ≈ 0.0783

3. P(0.25 < X < 0.40):

Calculate the z-scores of the boundaries:

z1 = (0.25 - 0.30) / sqrt(0.3 * 0.7 / 100) ≈ -0.71

z2 = (0.40 - 0.30) / sqrt(0.3 * 0.7 / 100) ≈ 1.41

Using a standard normal distribution table or calculator, find the area between z = -0.71 and z = 1.41:

P(0.25 < X < 0.40) ≈ 0.8124 - 0.2611 ≈ 0.5513

4. P(X ≤ 0.27):

Calculate the z-score of X:

z = (0.27 - 0.30) / sqrt(0.3 * 0.7 / 100) ≈ -0.45

Using a standard normal distribution table or calculator, find the area to the left of z = -0.45:

P(X ≤ 0.27) ≈ 0.3228

The Complete Question

A population has a proportion equal to 0.30. Calculate the following probabilities with n = 100:

P(X ≤ 0.35)

P(X > 0.40)

P(0.25 < X < 0.40)

P(X ≤ 0.27)

Additionally, assume the population follows a normal distribution.