Question

In Exercises 6.9 and 6.10 , indicate whether the Central Limit Theorem applies so that the sample proportions follow a normal distribution. In each case below, is the sample size large enough so that the sample proportions follow a normal distribution? (a) \(n=500\) and \(p=0.1\) (b) \(n=25\) and \(p=0.5\) (c) \(n=30\) and \(p=0.2\) (d) \(n=100\) and \(p=0.92\)

Short answer

The Central Limit Theorem applies to scenarios (a) and (b), where both yielded more than 10 successes and failures. However, it does not apply to scenarios (c) and (d) as they do not meet the minimum success and failure criteria.

Step-by-step solution

Test Sample Size and Proportion for (a)

Multiply the sample size \(n=500\) and the probability \(p=0.1\). Calculate \(np = 500 * 0.1 = 50\). Multiply the sample size by \(1-p\) to get \(n(1-p) = 500(1-0.1) = 450\). Since both are greater than 10, the Central Limit Theorem applies and the distribution can be considered normal.

Test Sample Size and Proportion for (b)

Multiply the sample size \(n=25\) and the probability \(p=0.5\). Calculate \(np = 25 * 0.5 = 12.5\). Similarly, calculate \(n(1-p) = 25(1-0.5) = 12.5\). Since both are greater than 10, the Central Limit Theorem applies and the distribution can be considered normal.

Test Sample Size and Proportion for (c)

For \(n=30\) and \(p=0.2\), \(np = 30 * 0.2 = 6\) and \(n(1-p) = 30(1-0.2) = 24\). The first calculation resulted in a number less than 10, which means that the Central Limit Theorem does not apply and we cannot assume a normal distribution.

Test Sample Size and Proportion for (d)

For \(n=100\) and \(p=0.92\), calculate \(np = 100 * 0.92 = 92\) and \(n(1-p) = 100(1-0.92) = 8\). Again, since one of the results is less than 10, we cannot apply the Central Limit Theorem, and the distribution may not be normal.

Key concepts

Sample Proportions

Understanding sample proportions is essential for deciphering the success of an action in a larger population. Suppose we're interested in the proportion of voters in a city who support a certain candidate. By collecting data from a simple random sample of the population, we obtain the sample proportion. This value, designated as 'p-hat' (\( \bar{p} \bar{p} \bar{p} \bar{p} \bar{p} \bar{p} \bar{p} \bar{p} \bar{p} \)), acts as an estimate of the true population proportion 'p'.

When we gather data from various samples, the sample proportions themselves have their own distribution, which is centered around the true population proportion. The shape of this distribution depends on the sample size and the true population proportion. If certain conditions are met, the Central Limit Theorem helps us approximate this distribution with a normal distribution which is incredibly useful for making inferences about the population.

Normal Distribution

The normal distribution, often called the bell curve due to its shape, is a continuous probability distribution that is symmetrical around the mean. Many natural phenomena tend to exhibit a normal distribution, given a large enough sample size.

The Central Limit Theorem (CLT) states that the sampling distribution of the sample mean (or proportion) will approach a normal distribution as the sample size increases, regardless of the shape of the population distribution. This is a powerful tool because it allows statisticians to make predictions and inferences about population parameters using the normal distribution, even when the population itself is not normally distributed. However, for sample proportions, the rule of thumb is that both 'np' and 'n(1-p)' must be greater than 10 for the normal approximation to be reliable.

Sample Size

In the realm of statistics, sample size plays a pivotal role in the accuracy of estimations and predictions. The sample size affects the variability of estimates: generally, a larger sample reduces the margin of error and yields more precise estimates. However, increasing the sample size has diminishing returns; doubling the sample size does not halve the margin of error but rather reduces it by a factor of the square root of two.

The Central Limit Theorem also hinges on 'sufficiently large' sample sizes to ensure normal distribution of sample means or proportions. The definition of 'sufficient' can vary, but a common benchmark is that a sample size is large enough if 'np' and 'n(1-p)' are both greater than 10. This rule helps to determine if the CLT can be applied to approximate a normal distribution for sample proportions, further supporting accurate probabilistic modeling and hypothesis testing.

Probability

Probability is the mathematical language we use to quantify uncertainty. It measures the likelihood of an event occurring, and it ranges from 0 (impossible event) to 1 (certain event). Understanding probabilities allows us to make informed decisions based on potential risks and benefits.

In statistics, the probability of an event can be estimated using the relative frequency of outcomes from a well-defined experimental procedure or random sample. With respect to the Central Limit Theorem, probability plays a crucial role in determining how 'unusual' a sample statistic is when compared to a population parameter. For instance, we can calculate the probability that a sample proportion falls within a certain range of the true population proportion. Utilizing the normal distribution, which the CLT provides as an approximation method, we can easily compute these probabilities using standard normal distribution tables or software, allowing us to test hypotheses and make predictions with more confidence.