Question

Flash bulbs manufactured by a certain company are sometimes defective. a. If \(5 \%\) of all such bulbs are defective, could the techniques of this section be used to approximate the probability that at least 5 of the bulbs in a random sample of size 50 are defective? If so, calculate this probability; if not, explain why not. b. Reconsider the question posed in Part (a) for the probability that at least 20 bulbs in a random sample of size 500 are defective.

Short answer

a. This approximation is not valid for a sample size of 50. b. The probability that at least 20 bulbs being defective in a sample size of 500 is 0.8413.

Step-by-step solution

Checking Validity of Approximation

The techniques of this section can be used to approximate the binomial probability if both np and n(1-p) are greater than or equal to 5. Let's check this for a random sample of size 50 and defective rate of \(5%\). Here, n=50, p=0.05. So, np = 50*0.05 = 2.5 and n(1-p) = 50*0.95 = 47.5. Since np < 5, the approximation is not valid in this case.

Calculation of Probability

As the approximation is not valid, we cannot calculate the probability using the techniques of this section for the first part of the question.

Rechecking Validity of Approximation

For a sample size of 500, let's re-apply the process: n=500, p=0.05. Again, np = 500*0.05 = 25 and n(1-p) = 500*0.95 = 475 which are both greater than or equal to 5. Hence, the approximation is valid in this case.

Calculate the Probability for larger sample size

We use the normal approximation to the binomial. Here, we calculate the z score using the following formula: \(z = (X - np) / \sqrt{np * (1-p)}\). Plugging in the numbers, we get: \(z = (20 - 25) / \sqrt{25*0.95} = -1\). Using standard normal distribution table, the probability of z being less than -1 is 0.1587, but we need the probability that z is greater than or equals to -1, which is \(1 - 0.1587 = 0.8413\).

Key concepts

Normal Approximation to the Binomial

Understanding when and how to apply the normal approximation to the binomial distribution is crucial in statistics. The binomial distribution describes the number of successes in a fixed number of independent trials, each with the same probability of success. For larger sample sizes, the binomial distribution can be approximated by the normal distribution, due to the Central Limit Theorem, which states that the sampling distribution of the sample means will tend to be normal, regardless of the shape of the population distribution.

This simplification is possible if certain conditions are met: both the expected number of successes, denoted by the term \(np\) and the expected number of failures, \(n(1-p)\), must be greater than or equal to 5. If these conditions are not met, as in the exercise where \(np<5\), the approximation is not considered valid, and a different method, such as the exact binomial calculation, needs to be used.

Probability Calculations

The realm of probability calculations allows us to quantify the chances of various outcomes. In the context of the binomial distribution, the probability of observing a certain number of successes in a set number of trials can be calculated exactly using the binomial formula. However, for large sample sizes or for ease of calculation, we often use the normal approximation to perform these calculations faster while maintaining reasonable accuracy.

When the normal approximation is applicable, the probability of having at least a certain number of successes (or failures) can be calculated by finding the area under the standard normal curve corresponding to the calculated z-score value. This translates a discrete probability problem into a continuous one, where the integral of the density function over the desired range gives us the probability we are seeking.

Validity of Statistical Approximations

The validity of statistical approximations like the normal approximation to the binomial distribution depend on adherence to key thresholds. As mentioned, the product of the number of trials and the probability of success/failure (\(np\) and \(n(1-p)\)) should both be greater than or equal to 5 for the normal approximation to be valid.

If these conditions are not met, the approximation could introduce significant errors in the calculations. Additionally, the shape of the binomial distribution should be reasonably symmetric, which happens when the probability of success is not too close to 0 or 1. Following these guidelines ensures that the approximation does not lead to misleading results and maintains the integrity of the statistical analysis.

Sampling Distribution

In statistics, the sampling distribution is a probability distribution of a statistic that is formed by considering all possible samples of a given size from a population. It is foundational to the field as it allows us to make inferences about population parameters.

Under the Central Limit Theorem, the sampling distribution of the sample proportion tends to be normally distributed when the sample size is large enough, even if the underlying population distribution is not normal. This is what justifies using the normal approximation for binomial probabilities in large samples. The sampling distribution's mean is the population mean, and its standard deviation, known as the standard error, depends on both the population standard deviation and the sample size.

Z-Score Calculation

A z-score is a statistical measure that describes a value's relationship to the mean of a group of values, expressed in terms of standard deviations away from the mean. Z-score calculation is pivotal for using the normal approximation in binomial probability.

The formula for calculating a z-score in the context of normal approximation to the binomial is \( z = \frac{X - np}{\sqrt{np(1-p)}} \), where \(X\) is the number of successes, \(n\) is the number of trials, and \(p\) is the probability of success on a given trial. By converting to a z-score, we can use standard normal distribution tables or software to find probabilities, which adds to our ability to analyze and understand probabilistic events with greater ease and accuracy.