Problem 12
Question
Suppose that \(H_{0}: \mu_{X}=\mu_{Y}\) is being tested against \(H_{1}\) : \(\mu_{X} \neq \mu_{Y}\), where \(\sigma_{X}^{2}\) and \(\sigma_{Y}^{2}\) are known to be \(17.6\) and \(22.9\), respectively. If \(n=10, m=20, \bar{x}=81.6\), and \(\bar{y}=79.9\), what \(P\)-value would be associated with the observed \(Z\) ratio?
Step-by-Step Solution
Verified Answer
To find the exact P-value, we firstly need to compute the standard error and Z statistic as directed in steps 1 and 2 above. Subsequently we would use scipy.stats.norm.sf() function in Python or standard normal distribution lookup table to find the P-value by locating the absolute value of our Z-score.
1Step 1: Calculate the standard error
The first step is to calculate the combined standard error of the two population means. The standard error can be calculated using the formula \(\sqrt{\( \sigma_{X}^{2} / n + \sigma_{Y}^{2} / m\) }\) where \( \sigma_{X}^{2} \) and \( \sigma_{Y}^{2} \) are the variance of population X and population Y, n is the sample size of population X, and m is the size of population Y. Plug in the values to get: \(\sqrt{\( 17.6 / 10 + 22.9 / 20\) }\).
2Step 2: Calculate the Z ratio
After getting the standard error, we can calculate the Z ratio or Z statistic. The formula for the Z statistic is \( (\bar{x}-\bar{y}) / SE \) where \(\bar{x}\) and \(\bar{y}\) are the sample means, and SE is the standard error calculated in step 1. So, the Z statistic can be calculated as \( (81.6 - 79.9) / standard_error \).
3Step 3: Calculate the P-value
The last step is to find the P-value. A P-value is a measure of the probability that an observed difference could have occurred just by random chance. The P-value can be found from the Z value using Z table or standard normal distribution table. If the calculated Z statistic value is Z, then we have two tail areas, because we are testing for unequality. Thus, the P-value is equal to two multiplied by the area that is greater than the absolute value of Z in the standard normal distribution. In Python, you could use scipy.stats.norm.sf(abs(Z score)) * 2 to calculate the P-value.
Key Concepts
Standard Error CalculationZ StatisticP-value ComputationSample Means ComparisonStandard Normal Distribution
Standard Error Calculation
When comparing two sample means to determine if they are significantly different from one another, a crucial step is to compute the standard error (SE) of their difference. The standard error acts as a gauge of how much we would expect our sample statistic, such as the mean, to vary if we took multiple samples from the same population.
In the context of our exercise, we are looking at the standard error of the difference between two population means, given by the formula: \[ SE = \sqrt{\frac{\sigma_{X}^2}{n} + \frac{\sigma_{Y}^2}{m}} \].
Here, \( \sigma_X^2 \) and \( \sigma_Y^2 \) are the variances of the two populations, while \( n \) and \( m \) are the sample sizes from each population. Filling in the values provided in our problem, we calculate the standard error of the difference between the sample means, which lays the foundation for the next steps in hypothesis testing.
In the context of our exercise, we are looking at the standard error of the difference between two population means, given by the formula: \[ SE = \sqrt{\frac{\sigma_{X}^2}{n} + \frac{\sigma_{Y}^2}{m}} \].
Here, \( \sigma_X^2 \) and \( \sigma_Y^2 \) are the variances of the two populations, while \( n \) and \( m \) are the sample sizes from each population. Filling in the values provided in our problem, we calculate the standard error of the difference between the sample means, which lays the foundation for the next steps in hypothesis testing.
Z Statistic
The Z statistic, also known as a Z score, is a key component in hypothesis testing when working with large sample sizes or known population variances. This value tells us how many standard errors away our sample mean is from the null hypothesis value.
The formula for the Z statistic when comparing two means is: \[ Z = \frac{\bar{x} - \bar{y}}{SE} \], where \( \bar{x} \) and \( \bar{y} \) are the sample means of group X and Y.
By substituting our calculated standard error and the given sample means into this formula, we receive a Z statistic which we will then use to assess the probability of observing such a difference in means under the null hypothesis. This step transitions us to compute the P-value.
The formula for the Z statistic when comparing two means is: \[ Z = \frac{\bar{x} - \bar{y}}{SE} \], where \( \bar{x} \) and \( \bar{y} \) are the sample means of group X and Y.
By substituting our calculated standard error and the given sample means into this formula, we receive a Z statistic which we will then use to assess the probability of observing such a difference in means under the null hypothesis. This step transitions us to compute the P-value.
P-value Computation
The P-value is a powerful statistic that helps researchers determine the significance of their results. It represents the probability of observing a test statistic at least as extreme as the one computed, assuming the null hypothesis is true.
To find the P-value from our Z statistic, we look at the tail areas of the standard normal distribution that lie beyond the observed Z value. Since our hypothesis test is two-tailed (\( \mu_X eq \mu_Y \)), we consider both ends of the distribution. The P-value is calculated as the sum of these areas, or more traditionally, as twice the area beyond the observed Z value's absolute magnitude.
Using standard normal distribution tables, or software tools like Python's scipy.stats library, we can accurately determine the P-value associated with our Z statistic, which will guide us towards making a decision about the null hypothesis.
To find the P-value from our Z statistic, we look at the tail areas of the standard normal distribution that lie beyond the observed Z value. Since our hypothesis test is two-tailed (\( \mu_X eq \mu_Y \)), we consider both ends of the distribution. The P-value is calculated as the sum of these areas, or more traditionally, as twice the area beyond the observed Z value's absolute magnitude.
Using standard normal distribution tables, or software tools like Python's scipy.stats library, we can accurately determine the P-value associated with our Z statistic, which will guide us towards making a decision about the null hypothesis.
Sample Means Comparison
Comparing sample means is a fundamental technique to infer differences between population parameters. In our exercise, we are tasked with comparing the means of two independent samples to see if they suggest a significant difference between the population means.
To perform this comparison, we calculate the difference between the sample means, then assess whether this observed difference is notably greater than what could be attributed to random chance alone. The SE and Z statistic, previously computed, are instrumental in this process. They enable us to standardize the difference in means and relate it to known probability distributions, thereby deriving the P-value as an objective measure of significance.
To perform this comparison, we calculate the difference between the sample means, then assess whether this observed difference is notably greater than what could be attributed to random chance alone. The SE and Z statistic, previously computed, are instrumental in this process. They enable us to standardize the difference in means and relate it to known probability distributions, thereby deriving the P-value as an objective measure of significance.
Standard Normal Distribution
Understanding the standard normal distribution is essential for hypothesis testing, particularly when calculating the Z statistic and the corresponding P-value. The standard normal distribution is a special case of the normal distribution that has a mean of zero and a standard deviation of one. It's a key concept in statistics because it allows us to determine probabilities associated with Z scores.
When comparing our Z statistic to this distribution, we can find the likelihood of observing a sample mean at least as extreme as ours, if no true difference exists between the population means. Tools such as the Z-table map Z scores to probabilities, which tell us the percentage of data falling beyond a specific point in the standard normal distribution. This is how we ultimately calculate the P-value for our hypothesis test.
When comparing our Z statistic to this distribution, we can find the likelihood of observing a sample mean at least as extreme as ours, if no true difference exists between the population means. Tools such as the Z-table map Z scores to probabilities, which tell us the percentage of data falling beyond a specific point in the standard normal distribution. This is how we ultimately calculate the P-value for our hypothesis test.
Other exercises in this chapter
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