Final answer:
The normal distribution can approximate the binomial distribution if the sample size is large, and both np and nq are greater than 5. For a large n with p = 0.53, the normal approximation is appropriate, and probabilities can be calculated using z-scores, which are interpreted in the context of the empirical rule. The central limit theorem ensures that the sample means will be normally distributed with a large enough sample size.
Step-by-step explanation:
To decide whether you can use the normal distribution to approximate the binomial distribution, you have to check whether the sample size (n) is large enough and if the success probability (p) meets certain conditions. Specifically, the sample size should generally be greater than 30, and both np and nq (where q = 1 - p) must be greater than 5. These conditions ensure that the binomial distribution is sufficiently close to a normal curve for the approximation to be accurate.
For the given example with n = 300 and p = 0.53, because both 300(0.53) and 300(1 - 0.53) are greater than 5, it's appropriate to use the normal approximation. The expected mean (μ) of the binomial distribution would be np and the standard deviation (σ) would be the square root of npq. The probability questions such as the probability of the sample mean being greater than 100 would be calculated using the z-score from the normal distribution. A z-score of -2 would indicate that the sample mean is 2 standard deviations below the population mean, which is significant as per the empirical rule.
The Central Limit Theorem for Sample Means (Averages)
The central limit theorem states that the distribution of sample means will be approximately normal, regardless of the shape of the population distribution, provided the sample size is sufficiently large. This theorem underlies the principle allowing for the normal approximation to the binomial in this context.
In conclusion, a bell-shaped normal curve is an excellent approximation for binomial distributions in many cases, notably in situations involving large sample sizes and certain probabilities, and is a fundamental aspect of statistical analysis across various disciplines.