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Magazine Chapter 7: Problem 45
Determine whether a normal sampling distribution can be used to approximate the binomial distribution. If it can, test the claim. Claim: \(p=0.15 ; \alpha=0.05\) Sample statistics: \(\hat{p}=0.09, n=40\)
Short Answer
Expert verified
Yes, a normal approximation is valid; do not reject the claim that \(p = 0.15\).
Step by step solution
01
Check Normal Approximation Criteria
To determine whether a normal approximation can be used for the binomial distribution, we need to check two conditions: \(np \geq 5\) and \(n(1-p) \geq 5\). Given \(p = 0.15\) and \(n = 40\), calculate \(np = 40 \times 0.15 = 6\) and \(n(1-p) = 40 \times 0.85 = 34\). Both values satisfy the criteria.
02
Calculate the Test Statistic
Since both conditions for normal approximation are satisfied, we proceed with the test. The test statistic for a binomial distribution approximated by a normal distribution is \(z = \frac{\hat{p} - p}{\sqrt{\frac{p(1-p)}{n}}}\). Substitute the values: \(z = \frac{0.09 - 0.15}{\sqrt{\frac{0.15 \times 0.85}{40}}}\).
03
Simplify the Test Statistic
Calculate further: First, \(\sqrt{\frac{0.15 \times 0.85}{40}} = \sqrt{\frac{0.1275}{40}} = \sqrt{0.0031875} \approx 0.0564\). Second, \(z = \frac{-0.06}{0.0564} 鈮� -1.0641\).
04
Determine the Critical Value
For a significance level \(\alpha = 0.05\) (two-tailed test), the critical z-value is approximately 卤1.96. Compare the test statistic \(z = -1.0641\) to the critical z-value.
05
Make a Decision
Since \(-1.96 < -1.0641 < 1.96\), the test statistic does not fall into the critical region. Therefore, we do not reject the null hypothesis.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Binomial Distribution
The binomial distribution is a probability distribution that summarizes the likelihood of a value taking one of two independent states. Typically, this is referred to as "success" and "failure" for a binary experiment that is repeated multiple times. For each trial, the probability of success remains constant. For example, when flipping a coin, each flip represents a trial, where the probability of getting heads can be considered a success. If we flip the coin several times, the number of times we get heads follows a binomial distribution.
- The binomial distribution requires two parameters: the number of trials ( egin{align*} n ext{ ootnotesize)} and probability of success ( egin{align*} p ext{ ootnotesize)} for each trial.
- The outcomes of the binomial distribution are discrete and can be represented as 0, 1, 2, ..., up to the maximum number of trials.
- Sometimes, so long as certain conditions are met, we can approximate this distribution with a normal distribution to simplify the calculations, especially when dealing with larger sample sizes.
Normal Distribution
The normal distribution, often known as the bell curve due to its shape, is a continuous probability distribution. It's characterized by its symmetric shape where the mean, median, and mode all fall at the center of the distribution. In many real-world scenarios, data tends to be distributed normally, which allows statisticians to make inferences and predictions about a population from sample data.
- The normal distribution is defined by two key parameters: the mean (\( \mu \)) and the standard deviation (\( \sigma \)).
- Key properties include being symmetric around the mean, with about 68% of data falling within one standard deviation, 95% within two, and 99.7% within three standard deviations.
- To approximate a binomial distribution with a normal distribution, the criteria \( np \geq 5\) and \( n(1-p) \geq 5\) need to be satisfied. This approximation is useful in hypothesis testing.
Hypothesis Testing
Hypothesis testing is a statistical method that allows us to make decisions or inferences about a population based on sample data. It starts with a claim or hypothesis about a population parameter, which is then tested using statistical calculations.
- The two hypotheses involved are:
- Null hypothesis (\( H_0 \)): a statement of no effect or no difference, which we want to test against.
- Alternative hypothesis (\( H_1 \) or \( H_a \)): a statement that indicates some effect or difference.
- The process involves calculating a test statistic and comparing it to a critical value, which is determined by the chosen significance level (\( \alpha \)). If the test statistic falls into the critical region, \( H_0 \) is rejected.
- The significance level (\( \alpha \)) reflects the probability of incorrectly rejecting the null hypothesis. Commonly used \( \alpha \) values are 0.05, 0.01, and 0.10.
Test Statistic
A test statistic is a standardized value that is calculated from sample data during a hypothesis test. It determines how far the data is from the null hypothesis. In cases where normal approximation is used for a binomial distribution, the test statistic is often calculated using the z-score formula:
- \[ z = \frac{\hat{p} - p}{\sqrt{\frac{p(1-p)}{n}}} \]
- In the formula, \( \hat{p} \) is the sample proportion, \( p \) is the claimed population proportion, and \( n \) is the sample size.
- The z-score tells us the number of standard deviations the sample proportion is from the population proportion. A high absolute value of the z-score might suggest a significant difference between the sample and population proportions.
- In comparison to a critical value derived from the normal distribution and significance level, the z-score helps in deciding whether to reject the null hypothesis.
