Question

A random sample of size 100 is taken from a population described by

the proportion p = 0.60.
The probability that the sample proportion is between 0.55 and 0.62 is

Answer 1

The probability that the sample proportion is between 0.55 and 0.62 is approximately 0.3034.

How to find the probability

To find the probability that the sample proportion is between 0.55 and 0.62, use the normal approximation to the binomial distribution.

Given that the sample size is large (n = 100) and the proportion p = 0.60, we can approximate the distribution of the sample proportion using the normal distribution.

Step 1: Calculate the mean and standard deviation of the sample proportion.

The mean of the sample proportion is equal to the population proportion: μ = p = 0.60.

The standard deviation of the sample proportion is calculated as:

σ =
`\sqrt`((p * (1 - p)) / n)

=
`\sqrt`((0.60 * 0.40) / 100)

=
`\sqrt`(0.0024) ≈ 0.049.

Step 2: Standardize the values.

To use the standard normal distribution, standardize the values of 0.55 and 0.62.

For 0.55:

Z1 = (0.55 - μ) / σ = (0.55 - 0.60) / 0.049 ≈ -1.02

For 0.62:

Z2 = (0.62 - μ) / σ = (0.62 - 0.60) / 0.049 ≈ 0.41

Step 3: Find the probability.

Now find the probability using the standard normal distribution table or a calculator.

P(0.55 < p < 0.62) = P(-1.02 < Z < 0.41)

Using the standard normal distribution table or a calculator, we can find the probability associated with each Z-value and subtract the cumulative probabilities.

P(-1.02 < Z < 0.41) ≈ 0.6554 - 0.3520 ≈ 0.3034

Therefore, the probability that the sample proportion is between 0.55 and 0.62 is approximately 0.3034.