A Missing Data Value From A Set Of Data Has A $z$-score Of $-2.1$. Fred Already Calculated The Mean And Standard Deviation To Be $\mu=43$ And $\sigma=2$. What Was The Missing Data Value? Round The Answer To The

Understanding Z-Scores and Their Significance

In statistics, 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 z-score formula is given by:

${ z = \frac{X - \mu}{\sigma} }$

where:

• $X$ is the value of the element,
• $\mu$ is the mean of the dataset, and
• $\sigma$ is the standard deviation of the dataset.

Calculating the Missing Data Value

Given that the z-score of the missing data value is $-2.1$, the mean is $\mu=43$, and the standard deviation is $\sigma=2$, we can use the z-score formula to find the missing data value.

We are given that the z-score is $-2.1$, so we can set up the equation as follows:

${ -2.1 = \frac{X - 43}{2} }$

Solving for the Missing Data Value

To solve for the missing data value, we need to isolate $X$ on one side of the equation. We can do this by multiplying both sides of the equation by $2$ to get rid of the fraction.

${ -2.1 \times 2 = X - 43 }$

${ -4.2 = X - 43 }$

Next, we can add $43$ to both sides of the equation to get:

${ -4.2 + 43 = X }$

${ 38.8 = X }$

Conclusion

Therefore, the missing data value is $38.8$. This is the value that would result in a z-score of $-2.1$ given the mean and standard deviation of the dataset.

Rounding the Answer

The problem asks us to round the answer to the nearest tenth. Since the missing data value is $38.8$, we can round it to $38.8$.

Final Answer

The final answer is $\boxed{38.8}$.

Understanding the Significance of Z-Scores

Z-scores are an important concept in statistics as they provide a way to compare the values of different datasets. By calculating the z-score of a value, we can determine how many standard deviations away from the mean it is. This can be useful in a variety of applications, such as quality control, where we may want to identify values that are significantly different from the mean.

Using Z-Scores in Real-World Applications

Z-scores have many real-world applications, including:

• Quality Control: Z-scores can be used to identify values that are significantly different from the mean, which can indicate a problem with the manufacturing process.
• Finance: Z-scores can be used to evaluate the creditworthiness of a borrower by comparing their credit score to the mean credit score of a group.
• Medicine: Z-scores can be used to evaluate the effectiveness of a treatment by comparing the results to the mean results of a group.

Conclusion

In conclusion, z-scores are an important concept in statistics that provide a way to compare the values of different datasets. By calculating the z-score of a value, we can determine how many standard deviations away from the mean it is. This can be useful in a variety of applications, such as quality control, finance, and medicine.

Frequently Asked Questions

Q: What is a z-score, and how is it calculated?

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. The z-score formula is given by:

${ z = \frac{X - \mu}{\sigma} }$

where:

• $X$ is the value of the element,
• $\mu$ is the mean of the dataset, and
• $\sigma$ is the standard deviation of the dataset.

Q: How do I use the z-score formula to find the missing data value?

A: To use the z-score formula to find the missing data value, you need to rearrange the formula to isolate $X$. This can be done by multiplying both sides of the equation by $\sigma$ and then adding $\mu$ to both sides.

Q: What if I don't know the standard deviation of the dataset?

A: If you don't know the standard deviation of the dataset, you can't use the z-score formula to find the missing data value. However, you can use other methods, such as the mean absolute deviation (MAD) or the interquartile range (IQR), to estimate the standard deviation.

Q: Can I use the z-score formula to find the missing data value if the z-score is positive?

A: Yes, you can use the z-score formula to find the missing data value if the z-score is positive. The formula remains the same, and you can follow the same steps to isolate $X$.

Q: What if the z-score is zero?

A: If the z-score is zero, it means that the value is equal to the mean. In this case, you can't use the z-score formula to find the missing data value, as the value is already known.

Q: Can I use the z-score formula to find the missing data value if the dataset is not normally distributed?

A: The z-score formula assumes that the dataset is normally distributed. If the dataset is not normally distributed, you may need to use other methods, such as the median absolute deviation (MAD) or the interquartile range (IQR), to estimate the standard deviation.

Q: What if I have multiple missing data values?

A: If you have multiple missing data values, you can use the z-score formula to find each value individually. However, you may need to use other methods, such as multiple imputation, to estimate the missing values.

Q: Can I use the z-score formula to find the missing data value if the dataset has outliers?

A: The z-score formula assumes that the dataset does not have outliers. If the dataset has outliers, you may need to use other methods, such as the median absolute deviation (MAD) or the interquartile range (IQR), to estimate the standard deviation.

Q: What if I have a large dataset with many missing data values?

A: If you have a large dataset with many missing data values, you may need to use other methods, such as multiple imputation, to estimate the missing values. The z-score formula may not be suitable for large datasets with many missing values.

Conclusion

In conclusion, the z-score formula is a useful tool for finding missing data values. However, it assumes that the dataset is normally distributed and does not have outliers. If the dataset is not normally distributed or has outliers, you may need to use other methods to estimate the standard deviation. Additionally, if you have a large dataset with many missing data values, you may need to use other methods, such as multiple imputation, to estimate the missing values.