Table of contents

1. Intro to Stats and Collecting Data55m
2. Describing Data with Tables and Graphs1h 55m
3. Describing Data Numerically1h 45m
4. Probability2h 16m
5. Binomial Distribution & Discrete Random Variables2h 33m
6. Normal Distribution and Continuous Random Variables1h 38m
7. Sampling Distributions & Confidence Intervals: Mean1h 53m
8. Sampling Distributions & Confidence Intervals: Proportion1h 12m
- Sampling Distribution of Sample Proportion
  29m
- Confidence Intervals for Population Proportion
  42m
9. Hypothesis Testing for One Sample2h 19m
10. Hypothesis Testing for Two Samples3h 22m
11. Correlation1h 6m
12. Regression1h 4m
13. Chi-Square Tests & Goodness of Fit1h 20m
14. ANOVA1h 0m

6. Normal Distribution and Continuous Random Variables

Standard Normal Distribution

Statistics

6. Normal Distribution and Continuous Random Variables

Standard Normal Distribution

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2:11 minutes

Problem 5.5.12

Textbook Question

In Exercises 9–14, write the binomial probability in words. Then, use a continuity correction to convert the binomial probability to a normal distribution probability.

P(x > 65)

Verified step by step guidance

Step 1: Write the binomial probability in words. The problem asks for the probability that the number of successes (x) is greater than 65 in a binomial distribution.

Step 2: Recall that a binomial distribution can be approximated by a normal distribution if the sample size is large enough (n is large, and both np and n(1-p) are greater than 5). This is known as the normal approximation to the binomial distribution.

Step 3: Identify the mean (μ) and standard deviation (σ) of the binomial distribution. The mean is given by μ = n * p, and the standard deviation is given by σ = √(n * p * (1 - p)), where n is the number of trials and p is the probability of success.

Step 4: Apply the continuity correction. Since the problem asks for P(x > 65), we adjust the inequality to P(x ≥ 65.5) to account for the discrete-to-continuous transition.

Step 5: Standardize the value using the z-score formula for the normal distribution: z = (x - μ) / σ. Substitute x = 65.5, μ, and σ into the formula to calculate the z-score. Then, use the standard normal distribution table or a calculator to find the probability corresponding to the z-score.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Binomial Probability

Binomial probability refers to the likelihood of obtaining a fixed number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. It is calculated using the binomial formula, which incorporates the number of trials, the number of successes, and the probability of success on each trial. This concept is essential for understanding scenarios where outcomes are binary, such as success/failure or yes/no.

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 significant in statistics because many phenomena tend to approximate a normal distribution under certain conditions, particularly when the sample size is large. The normal distribution allows for easier calculations and interpretations of probabilities compared to discrete distributions like the binomial.

Continuity Correction

Continuity correction is a technique used when approximating a discrete probability distribution, such as the binomial distribution, with a continuous distribution, like the normal distribution. This correction involves adjusting the discrete value by 0.5 units to account for the fact that the normal distribution is continuous. For example, to find P(x > 65) in a binomial context, one would calculate P(x > 65.5) in the normal approximation.

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Related Practice

Textbook Question

In Exercises 27–32, the random variable x is normally distributed with mean mu=74 and standard deviation sigma=8. Find the indicated probability.

P(72 < x < 82)

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Textbook Question

In Exercises 1–4, the sample size n, probability of success p, and probability of failure q are given for a binomial experiment. Determine whether you can use a normal distribution to approximate the distribution of x.

n=18, p=0.90, q=0.10

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Textbook Question

In Exercises 5–8, match the binomial probability statement with its corresponding normal distribution probability statement (a)–(d) after a continuity correction.

P(x≥109)

a. P(x>109.5)

b. P(x<108.5)

c. P(x<109.5)

d. P(x>108.5)

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Textbook Question

In Exercises 9–14, write the binomial probability in words. Then, use a continuity correction to convert the binomial probability to a normal distribution probability.

P(x ≥ 110)

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Textbook Question

In Exercises 9–14, write the binomial probability in words. Then, use a continuity correction to convert the binomial probability to a normal distribution probability.

P(x ≤ 150)

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Textbook Question

In Exercises 9–14, write the binomial probability in words. Then, use a continuity correction to convert the binomial probability to a normal distribution probability.

P(55 < x < 60)

n=24, p=0.85, q=0.15

n=20, p=0.65, q=0.35