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Answer :
Sure! Let's break down the solution into clear, step-by-step explanations:
### 1. Data set with a mean of 100 and a standard deviation of 15
To solve these questions, we'll use the properties of a normal distribution. The normal distribution is a bell-shaped curve that is symmetrical around its mean. The percent of data falling between any two values can be found by using the concept of z-scores, which measure how many standard deviations a data point is from the mean.
#### a. Percent of data between 85 and 115
1. Calculate z-scores:
- For 85: [tex]\( z = \frac{85 - 100}{15} \)[/tex]
- For 115: [tex]\( z = \frac{115 - 100}{15} \)[/tex]
2. Find the percent of data between these z-scores:
- Use a standard normal distribution table or a calculator to determine the percentage of data between the calculated z-scores.
The result is approximately 68.27%.
#### b. Percent of data between 70 and 100
1. Calculate z-scores:
- For 70: [tex]\( z = \frac{70 - 100}{15} \)[/tex]
- For 100: [tex]\( z = \frac{100 - 100}{15} \)[/tex] (which is 0)
2. Find the percent of data between these z-scores:
- Use a standard normal distribution table or a calculator.
The result is approximately 47.72%.
#### c. Percent of data between 85 and 145
1. Calculate z-scores:
- For 85: [tex]\( z = \frac{85 - 100}{15} \)[/tex]
- For 145: [tex]\( z = \frac{145 - 100}{15} \)[/tex]
2. Find the percent of data between these z-scores:
- Use a standard normal distribution table or a calculator.
The result is approximately 84.00%.
### 2. Normal distribution with a mean of 50 and a standard deviation of 7
To find the z-score for a data value:
1. Calculate the z-score for 60:
[tex]\[
z = \frac{60 - 50}{7} \approx 1.43
\][/tex]
### 3. Skewed left data set
- If a data set is skewed left, that means the tail on the left side of the distribution is longer than the right side. In such cases, the mean is affected by the lower values more than the median is.
- Recommendation: Use the median as the measure of central tendency because it's less affected by the skewness and provides a better representation of the central location within the data.
I hope this helps you understand how these calculations are done! If you have further questions, feel free to ask.
### 1. Data set with a mean of 100 and a standard deviation of 15
To solve these questions, we'll use the properties of a normal distribution. The normal distribution is a bell-shaped curve that is symmetrical around its mean. The percent of data falling between any two values can be found by using the concept of z-scores, which measure how many standard deviations a data point is from the mean.
#### a. Percent of data between 85 and 115
1. Calculate z-scores:
- For 85: [tex]\( z = \frac{85 - 100}{15} \)[/tex]
- For 115: [tex]\( z = \frac{115 - 100}{15} \)[/tex]
2. Find the percent of data between these z-scores:
- Use a standard normal distribution table or a calculator to determine the percentage of data between the calculated z-scores.
The result is approximately 68.27%.
#### b. Percent of data between 70 and 100
1. Calculate z-scores:
- For 70: [tex]\( z = \frac{70 - 100}{15} \)[/tex]
- For 100: [tex]\( z = \frac{100 - 100}{15} \)[/tex] (which is 0)
2. Find the percent of data between these z-scores:
- Use a standard normal distribution table or a calculator.
The result is approximately 47.72%.
#### c. Percent of data between 85 and 145
1. Calculate z-scores:
- For 85: [tex]\( z = \frac{85 - 100}{15} \)[/tex]
- For 145: [tex]\( z = \frac{145 - 100}{15} \)[/tex]
2. Find the percent of data between these z-scores:
- Use a standard normal distribution table or a calculator.
The result is approximately 84.00%.
### 2. Normal distribution with a mean of 50 and a standard deviation of 7
To find the z-score for a data value:
1. Calculate the z-score for 60:
[tex]\[
z = \frac{60 - 50}{7} \approx 1.43
\][/tex]
### 3. Skewed left data set
- If a data set is skewed left, that means the tail on the left side of the distribution is longer than the right side. In such cases, the mean is affected by the lower values more than the median is.
- Recommendation: Use the median as the measure of central tendency because it's less affected by the skewness and provides a better representation of the central location within the data.
I hope this helps you understand how these calculations are done! If you have further questions, feel free to ask.
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