Suppose that we have a population with proportion P=0.25 and...

Suppose that we have a population with proportion $P=0.25$ and a random sample of size $n=200$ drawn from the population. a. What is the probability that the sample proportion is greater than 0.31? b. What is the probability that the sample proportion is less than $0.14$ ? c. What is the probability that the sample proportion is between $0.24$ and $0.40$ ?

Suppose that we have a population with proportion $P=0.25$ and a random sample of size $n=200$ drawn from the population.
a. What is the probability that the sample proportion is greater than 0.31?
b. What is the probability that the sample proportion is less than $0.14$ ?
c. What is the probability that the sample proportion is between $0.24$ and $0.40$ ?

Key Concepts

Normal Approximation in Probability Calculations

With the sampling distribution approximated by a normal distribution thanks to the Central Limit Theorem, one can calculate probabilities regarding the sample proportion by using the normal probability table or software tools. This step involves finding the area under the normal curve corresponding to the z-scores derived from the observed values of the sample proportion.

Standardization (Z-score Transformation)

Standardization involves converting the sample proportion to a z-score by subtracting the mean (population proportion) and dividing by the standard error. This transformation allows one to utilize the standard normal distribution to find probabilities corresponding to different sample proportions.

Standard Error of the Sample Proportion

The standard error measures the spread or variability of the sampling distribution of the sample proportion. It is computed using the formula sqrt[p(1 - p)/n], where p is the population proportion and n is the sample size. This value is crucial for assessing how much the sample proportion is expected to deviate from the true population proportion.

Central Limit Theorem

The Central Limit Theorem states that, for a sufficiently large sample size, the sampling distribution of the sample proportion will be approximately normally distributed, regardless of the underlying distribution of the population. This approximation enables the use of normal probability methods to calculate probabilities about the sample proportion.

Sampling Distribution of a Sample Proportion

This concept refers to the probability distribution of the sample proportion, which indicates how the proportion calculated from a random sample varies from sample to sample. When samples are taken from a population with a known proportion, the sampling distribution models the variability of these sample proportions and underlies many inferential statistical techniques.

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