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

User Hovo
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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%.

User Rasika
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6 votes

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.

User Vladimir Obrizan
by
7.6k points
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