6. Normal Distribution and Continuous Random Variables
Standard Normal Distribution
1:54 minutesProblem 6.6.1c
Textbook 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
Verified step by step guidance1
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.
Verified video answer for a similar problem:This video solution was recommended by our tutors as helpful for the problem above
Video duration:
1mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Normal Distribution
The normal distribution is a continuous probability distribution characterized by its bell-shaped curve, defined by its mean and standard deviation. It is widely used in statistics because many phenomena tend to follow this distribution. In hypothesis testing, the normal distribution helps approximate the behavior of sample proportions, especially with large sample sizes, allowing for easier calculation of probabilities.
Recommended video:
Guided course
09:47
Finding Standard Normal Probabilities using z-Table
Continuity Correction
Continuity correction is a technique used when a discrete distribution is approximated by a continuous distribution, such as the normal distribution. It involves adjusting the discrete values by 0.5 to account for the fact that continuous distributions can take on any value within a range, while discrete distributions can only take specific values. This correction improves the accuracy of probability estimates, particularly in binomial distributions.
Recommended video:
06:23Using the Normal Distribution to Approximate Binomial Probabilities
Probability Calculation
Probability calculation involves determining the likelihood of a specific outcome occurring within a given set of conditions. In the context of the question, it refers to finding the area under the normal distribution curve that corresponds to the event of having more than 502 boys in a sample of 1000 births. This is done by calculating the z-score and using standard normal distribution tables or software to find the corresponding probabilities.
Recommended video:
Guided course
07:09Probability From Given Z-Scores - TI-84 (CE) Calculator
9:47mWatch next
Master Finding Standard Normal Probabilities using z-Table with a bite sized video explanation from Patrick
Start learning
