Indicate Which Student Has The Greater $z$ Score. Show Your Work.$\[ \begin{array}{l|l|l|l} \text{Math Major} & X=53 & \bar{X}=50.5 & S=5.2 \\ \hline \text{Statistics Major} & X=80 & \bar{X}=75.1 & S=7.3 \end{array} \\]A) Both
Indicate which student has the greater z score. Show your work.
A z score is a measure of how many standard deviations an element is from the mean. It is calculated by subtracting the mean from the value and then dividing by the standard deviation. The formula for calculating a z score is:
z = (X - μ) / σ
where:
- X is the value of the element
- μ is the mean of the dataset
- σ is the standard deviation of the dataset
We are given two students, a Math Major and a Statistics Major, with their respective values, means, and standard deviations. We need to calculate the z scores for both students and determine which one has the greater z score.
Math Major
| Value | Mean (μ) | Standard Deviation (σ) |
|---|---|---|
| 53 | 50.5 | 5.2 |
To calculate the z score for the Math Major, we use the formula:
z = (X - μ) / σ = (53 - 50.5) / 5.2 = 2.5 / 5.2 = 0.48
Statistics Major
| Value | Mean (μ) | Standard Deviation (σ) |
|---|---|---|
| 80 | 75.1 | 7.3 |
To calculate the z score for the Statistics Major, we use the formula:
z = (X - μ) / σ = (80 - 75.1) / 7.3 = 4.9 / 7.3 = 0.67
Now that we have calculated the z scores for both students, we can compare them to determine which one has the greater z score.
The Math Major has a z score of 0.48, while the Statistics Major has a z score of 0.67. Since 0.67 is greater than 0.48, the Statistics Major has the greater z score.
In conclusion, the Statistics Major has the greater z score. This means that the Statistics Major's value is more than 0.67 standard deviations away from the mean, while the Math Major's value is only 0.48 standard deviations away from the mean.
The z score is an important concept in statistics, as it allows us to compare values across different datasets. By calculating the z score, we can determine how many standard deviations away from the mean a value is, which can be useful in a variety of applications, such as hypothesis testing and confidence intervals.
In this case, the Statistics Major's greater z score indicates that their value is more extreme than the Math Major's value. This could be due to a variety of factors, such as differences in the datasets or the calculation of the z score.
The z score has a variety of real-world applications, including:
- Hypothesis testing: The z score can be used to test hypotheses about a population mean.
- Confidence intervals: The z score can be used to construct confidence intervals for a population mean.
- Data analysis: The z score can be used to analyze data and determine how many standard deviations away from the mean a value is.
Q: What is a z score?
A: A z score is a measure of how many standard deviations an element is from the mean. It is calculated by subtracting the mean from the value and then dividing by the standard deviation.
Q: How is a z score calculated?
A: The formula for calculating a z score is:
z = (X - μ) / σ
where:
- X is the value of the element
- μ is the mean of the dataset
- σ is the standard deviation of the dataset
Q: What does a z score of 0 mean?
A: A z score of 0 means that the value is equal to the mean. In other words, the value is not deviating from the mean at all.
Q: What does a positive z score mean?
A: A positive z score means that the value is greater than the mean. The larger the positive z score, the more the value is deviating from the mean.
Q: What does a negative z score mean?
A: A negative z score means that the value is less than the mean. The larger the negative z score, the more the value is deviating from the mean.
Q: How do I interpret a z score?
A: To interpret a z score, you need to understand what it means in the context of the problem. A z score of 1, for example, means that the value is 1 standard deviation away from the mean. A z score of 2 means that the value is 2 standard deviations away from the mean.
Q: Can I use z scores to compare values across different datasets?
A: Yes, you can use z scores to compare values across different datasets. Since z scores are standardized, they can be used to compare values that have different units or scales.
Q: What are some common applications of z scores?
A: Some common applications of z scores include:
- Hypothesis testing: The z score can be used to test hypotheses about a population mean.
- Confidence intervals: The z score can be used to construct confidence intervals for a population mean.
- Data analysis: The z score can be used to analyze data and determine how many standard deviations away from the mean a value is.
Q: Can I use z scores with non-normal data?
A: No, you should not use z scores with non-normal data. Z scores assume that the data is normally distributed, and using them with non-normal data can lead to incorrect conclusions.
Q: How do I calculate the z score for a grouped dataset?
A: To calculate the z score for a grouped dataset, you need to use the midpoint of the group as the value. The midpoint is the average of the lower and upper bounds of the group.
Q: Can I use z scores to compare values that have different units?
A: Yes, you can use z scores to compare values that have different units. Since z scores are standardized, they can be used to compare values that have different units or scales.
Q: What are some common mistakes to avoid when using z scores?
A: Some common mistakes to avoid when using z scores include:
- Assuming normality: Z scores assume that the data is normally distributed. If the data is not normally distributed, you should not use z scores.
- Using z scores with non-standardized data: Z scores should only be used with standardized data. If the data is not standardized, you should not use z scores.
- Interpreting z scores without context: Z scores should be interpreted in the context of the problem. Without context, z scores can be misleading.