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) \(n=80\) and \(p=0.1\) (b) \(n=25\) and \(p=0.8\) (c) \(n=50\) and \(p=0.4\) (d) \(n=200\) and \(p=0.7\)

Short answer

Yes, in all the given cases, i.e., (a), (b), (c) and (d), the sample size is large enough to apply the Central Limit Theorem and the sample proportions follow a normal distribution.

Step-by-step solution

Check Case (a)

Apply the rule \(np ≥ 5\) and \(n(1-p) ≥ 5\): For \(n=80\) and \(p=0.1\), calculate \(np = 80*0.1 = 8\) and \(n(1-p) = 80*(1-0.1) = 72\). Both values are greater than or equal to 5, so the Central Limit Theorem applies and the sample proportions follow a normal distribution in this case.

Check Case (b)

For \(n=25\) and \(p=0.8\), calculate \(np = 25*0.8 = 20\) and \(n(1-p) = 25*(1-0.8)= 5\). Both values are greater than or equal to 5, so the Central Limit Theorem applies here as well.

Check Case (c)

For \(n=50\) and \(p=0.4\), calculate \(np = 50*0.4 = 20\) and \(n(1-p) = 50*(1-0.4) = 30\). Both values are greater than or equal to 5, so the Central Limit Theorem applies in this case too.

Check Case (d)

For \(n=200\) and \(p=0.7\), calculate \(np = 200*0.7 = 140\) and \(n(1-p) = 200*(1-0.7) = 60\). Both values are greater than or equal to 5, so the Central Limit Theorem applies in this case as well.

Key concepts

Sample Proportions

When we perform a study or an experiment, often we want to know the proportion of times a certain outcome occurs within a sample. In statistical terms, this is referred to as a 'sample proportion'. For example, if we randomly select 100 people and measure how many of them prefer coffee over tea, the number preferring coffee divided by the total number of people in the sample is the sample proportion.

When dealing with sample proportions, it's important to determine whether they can be described using a normal distribution. The Central Limit Theorem (CLT) assists us in this determination. It tells us that, under certain conditions, the distribution of the sample proportions will approximate a normal distribution, regardless of the shape of the population distribution. These conditions generally relate to the sample size and the true population proportion.

Normal Distribution

The normal distribution, often referred to as the bell curve, is a fundamental concept in statistics because many phenomena naturally follow this pattern. Characteristics of a normal distribution include symmetry around the mean, with data tapering off equally on both sides, and a certain percentage of the data falling within one, two, or three standard deviations from the mean.

In the context of sample proportions, the normal distribution allows for simplifying assumptions when calculating probabilities and conducting hypothesis testing. The CLT helps in utilizing the normal distribution to make inferences about a population based on sample data, by confirming that the distribution of the sample means or proportions can be treated as if they are normal, given a sufficiently large sample size.

Sample Size

Sample size plays a crucial role in the application of the Central Limit Theorem. Intuitively, the larger the sample size, the more representative it is likely to be of the population. When a sample size is too small, the sample proportion may not accurately reflect the true population proportion.

In our exercise examples, we checked if sample sizes were large enough by using rules such as the so-called 'np rule', where both products, np and n(1-p), should be greater than or equal to 5. This rule ensures that the distribution of the sample proportion is sufficiently 'normal' for the CLT to apply. For example in case (a), with a sample size of 80 and a population proportion of 0.1, we found that the sample size is indeed large enough.

Statistical Significance

Statistical significance is a measure of the likelihood that a given result or finding is caused by something other than mere random chance. It is quantified using a 'p-value'. A smaller p-value indicates that there's stronger evidence in favor of a statistically significant effect. In the context of the normal distribution of sample proportions, statistical significance plays a role in hypothesis testing.

For example, if we are assessing the effectiveness of a new drug, we would compare the proportion of successful treatments in our sample to what we would expect based on previous data or a control group. If the sample proportion is significantly different from the expected proportion (with a small p-value), we may conclude that there's a statistically significant effect due to the new drug rather than due to random chance.