Question

Continuity Correction In testing the assumption that the probability of a baby boy is 0.512, a geneticist obtains a random sample of 1000 births and finds that 502 of them are boys. Using the continuity correction, describe the area under the graph of a normal distribution corresponding to the following. (For example, the area corresponding to “the probability of at least 502 boys” is this: the area to the right of 501.5.)

c. The probability of more than 502 boys

Accepted Answer

Step 1: Understand the concept of continuity correction. In a binomial distribution, when approximating probabilities using a normal distribution, a continuity correction is applied to account for the discrete nature of the binomial variable. This involves adjusting the value by ±0.5 depending on the inequality being considered. Step 2: Identify the inequality in the problem. The problem asks for the probability of 'more than 502 boys.' In terms of the normal distribution, this corresponds to the area to the right of 502.5 (adjusted using the continuity correction). Step 3: Calculate the mean (μ) and standard deviation (σ) of the binomial distribution. The mean is given by μ = n * p, where n is the sample size (1000) and p is the probability of a boy (0.512). The standard deviation is calculated as σ = √(n * p * (1 - p)). Step 4: Convert the adjusted value (502.5) into a z-score using the formula: z = (X - μ) / σ, where X is the adjusted value, μ is the mean, and σ is the standard deviation. This z-score represents the number of standard deviations the value is away from the mean. Step 5: Use the z-score to find the corresponding probability. Look up the z-score in a standard normal distribution table or use statistical software to find the area to the right of the z-score. This area represents the probability of more than 502 boys.

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Key Concepts

Normal Distribution

Continuity Correction

Probability Calculation