Question
June
483.19
497.99
488.56
486.25
482.05
472.68
472.08
454.26
444.95
482.37
493.37
498.72
485.52
484.78
474.02
July
489.2
491.34
494.02
459.61
466.18
481.59
477.5
484.81
490.06
488.97
492.63
484.35
484.99
484.85
439.49
(a) Find the population standard deviation and variance for the prices in June. Round the answers to at least three decimal places.
The population standard deviation for the prices in June is
.
The population variance for the prices in June is..
(b) Find the population standard deviation and variance for the prices in July. Round the answers to at least three decimal places.
The population standard deviation for the prices in July is
.
The population variance for the prices in July is
483.19
497.99
488.56
486.25
482.05
472.68
472.08
454.26
444.95
482.37
493.37
498.72
485.52
484.78
474.02
July
489.2
491.34
494.02
459.61
466.18
481.59
477.5
484.81
490.06
488.97
492.63
484.35
484.99
484.85
439.49
(a) Find the population standard deviation and variance for the prices in June. Round the answers to at least three decimal places.
The population standard deviation for the prices in June is
.
The population variance for the prices in June is..
(b) Find the population standard deviation and variance for the prices in July. Round the answers to at least three decimal places.
The population standard deviation for the prices in July is
.
The population variance for the prices in July is
Answers
GPT-4o mini
To calculate the population standard deviation and variance, we can follow these steps:
1. **Calculate the mean (average)** of the data.
2. **Calculate the variance** using the formula:
\[
\text{Variance} = \frac{\sum (x_i - \mu)^2}{N}
\]
where \(x_i\) are the data points, \(\mu\) is the mean, and \(N\) is the number of data points.
3. **Calculate the standard deviation** as the square root of the variance.
### (a) Prices in June
**Data:**
June prices:
483.19, 497.99, 488.56, 486.25, 482.05, 472.68, 472.08, 454.26, 444.95, 482.37, 493.37, 498.72, 485.52, 484.78, 474.02
**Step 1: Calculate the mean (μ):**
\[
\mu = \frac{\sum x_i}{N} = \frac{483.19 + 497.99 + 488.56 + 486.25 + 482.05 + 472.68 + 472.08 + 454.26 + 444.95 + 482.37 + 493.37 + 498.72 + 485.52 + 484.78 + 474.02}{15}
\]
\[
\mu = \frac{7297.77}{15} \approx 486.585
\]
**Step 2: Calculate the variance:**
\[
\text{Variance} = \frac{\sum (x_i - \mu)^2}{N}
\]
Calculating each \((x_i - \mu)^2\):
- \((483.19 - 486.585)^2 \approx 11.675\)
- \((497.99 - 486.585)^2 \approx 130.433\)
- \((488.56 - 486.585)^2 \approx 2.353\)
- \((486.25 - 486.585)^2 \approx 0.111\)
- \((482.05 - 486.585)^2 \approx 20.935\)
- \((472.68 - 486.585)^2 \approx 189.160\)
- \((472.08 - 486.585)^2 \approx 210.707\)
- \((454.26 - 486.585)^2 \approx 1026.169\)
- \((444.95 - 486.585)^2 \approx 1721.545\)
- \((482.37 - 486.585)^2 \approx 0.038\)
- \((493.37 - 486.585)^2 \approx 45.563\)
- \((498.72 - 486.585)^2 \approx 146.383\)
- \((485.52 - 486.585)^2 \approx 1.130\)
- \((484.78 - 486.585)^2 \approx 3.218\)
- \((474.02 - 486.585)^2 \approx 156.131\)
Now summing these values:
\[
\sum (x_i - \mu)^2 \approx 2415.272
\]
So,
\[
\text{Variance} \approx \frac{2415.272}{15} \approx 161.018
\]
**Standard deviation:**
\[
\text{Standard deviation} = \sqrt{\text{Variance}} \approx \sqrt{161.018} \approx 12.688
\]
### Final Answers for June:
- Population standard deviation: **12.688**
- Population variance: **161.018**
---
### (b) Prices in July
**Data:**
July prices:
489.2, 491.34, 494.02, 459.61, 466.18, 481.59, 477.5, 484.81, 490.06, 488.97, 492.63, 484.35, 484.99, 484.85, 439.49
**Step 1: Calculate the mean (μ):**
\[
\mu = \frac{\sum x_i}{N} = \frac{489.2 + 491.34 + 494.02 + 459.61 + 466.18 + 481.59 + 477.5 + 484.81 + 490.06 + 488.97 + 492.63 + 484.35 + 484.99 + 484.85 + 439.49}{15}
\]
\[
\mu = \frac{7253.43}{15} \approx 483.562
\]
**Step 2: Calculate the variance:**
Calculating \((x_i - \mu)^2\):
- \((489.2 - 483.562)^2 \approx 31.056\)
- \((491.34 - 483.562)^2 \approx 60.852\)
- \((494.02 - 483.562)^2 \approx 107.318\)
- \((459.61 - 483.562)^2 \approx 565.446\)
- \((466.18 - 483.562)^2 \approx 299.008\)
- \((481.59 - 483.562)^2 \approx 3.834\)
- \((477.5 - 483.562)^2 \approx 36.373\)
- \((484.81 - 483.562)^2 \approx 1.552\)
- \((490.06 - 483.562)^2 \approx 42.035\)
- \((488.97 - 483.562)^2 \approx 28.862\)
- \((492.63 - 483.562)^2 \approx 82.378\)
- \((484.35 - 483.562)^2 \approx 0.617\)
- \((484.99 - 483.562)^2 \approx 1.994\)
- \((484.85 - 483.562)^2 \approx 1.674\)
- \((439.49 - 483.562)^2 \approx 1917.823\)
Now summing these values:
\[
\sum (x_i - \mu)^2 \approx 2720.177
\]
So,
\[
\text{Variance} \approx \frac{2720.177}{15} \approx 181.345
\]
**Standard deviation:**
\[
\text{Standard deviation} = \sqrt{\text{Variance}} \approx \sqrt{181.345} \approx 13.451
\]
### Final Answers for July:
- Population standard deviation: **13.451**
- Population variance: **181.345**
Overall:
- June: Standard deviation = 12.688, Variance = 161.018
- July: Standard deviation = 13.451, Variance = 181.345
1. **Calculate the mean (average)** of the data.
2. **Calculate the variance** using the formula:
\[
\text{Variance} = \frac{\sum (x_i - \mu)^2}{N}
\]
where \(x_i\) are the data points, \(\mu\) is the mean, and \(N\) is the number of data points.
3. **Calculate the standard deviation** as the square root of the variance.
### (a) Prices in June
**Data:**
June prices:
483.19, 497.99, 488.56, 486.25, 482.05, 472.68, 472.08, 454.26, 444.95, 482.37, 493.37, 498.72, 485.52, 484.78, 474.02
**Step 1: Calculate the mean (μ):**
\[
\mu = \frac{\sum x_i}{N} = \frac{483.19 + 497.99 + 488.56 + 486.25 + 482.05 + 472.68 + 472.08 + 454.26 + 444.95 + 482.37 + 493.37 + 498.72 + 485.52 + 484.78 + 474.02}{15}
\]
\[
\mu = \frac{7297.77}{15} \approx 486.585
\]
**Step 2: Calculate the variance:**
\[
\text{Variance} = \frac{\sum (x_i - \mu)^2}{N}
\]
Calculating each \((x_i - \mu)^2\):
- \((483.19 - 486.585)^2 \approx 11.675\)
- \((497.99 - 486.585)^2 \approx 130.433\)
- \((488.56 - 486.585)^2 \approx 2.353\)
- \((486.25 - 486.585)^2 \approx 0.111\)
- \((482.05 - 486.585)^2 \approx 20.935\)
- \((472.68 - 486.585)^2 \approx 189.160\)
- \((472.08 - 486.585)^2 \approx 210.707\)
- \((454.26 - 486.585)^2 \approx 1026.169\)
- \((444.95 - 486.585)^2 \approx 1721.545\)
- \((482.37 - 486.585)^2 \approx 0.038\)
- \((493.37 - 486.585)^2 \approx 45.563\)
- \((498.72 - 486.585)^2 \approx 146.383\)
- \((485.52 - 486.585)^2 \approx 1.130\)
- \((484.78 - 486.585)^2 \approx 3.218\)
- \((474.02 - 486.585)^2 \approx 156.131\)
Now summing these values:
\[
\sum (x_i - \mu)^2 \approx 2415.272
\]
So,
\[
\text{Variance} \approx \frac{2415.272}{15} \approx 161.018
\]
**Standard deviation:**
\[
\text{Standard deviation} = \sqrt{\text{Variance}} \approx \sqrt{161.018} \approx 12.688
\]
### Final Answers for June:
- Population standard deviation: **12.688**
- Population variance: **161.018**
---
### (b) Prices in July
**Data:**
July prices:
489.2, 491.34, 494.02, 459.61, 466.18, 481.59, 477.5, 484.81, 490.06, 488.97, 492.63, 484.35, 484.99, 484.85, 439.49
**Step 1: Calculate the mean (μ):**
\[
\mu = \frac{\sum x_i}{N} = \frac{489.2 + 491.34 + 494.02 + 459.61 + 466.18 + 481.59 + 477.5 + 484.81 + 490.06 + 488.97 + 492.63 + 484.35 + 484.99 + 484.85 + 439.49}{15}
\]
\[
\mu = \frac{7253.43}{15} \approx 483.562
\]
**Step 2: Calculate the variance:**
Calculating \((x_i - \mu)^2\):
- \((489.2 - 483.562)^2 \approx 31.056\)
- \((491.34 - 483.562)^2 \approx 60.852\)
- \((494.02 - 483.562)^2 \approx 107.318\)
- \((459.61 - 483.562)^2 \approx 565.446\)
- \((466.18 - 483.562)^2 \approx 299.008\)
- \((481.59 - 483.562)^2 \approx 3.834\)
- \((477.5 - 483.562)^2 \approx 36.373\)
- \((484.81 - 483.562)^2 \approx 1.552\)
- \((490.06 - 483.562)^2 \approx 42.035\)
- \((488.97 - 483.562)^2 \approx 28.862\)
- \((492.63 - 483.562)^2 \approx 82.378\)
- \((484.35 - 483.562)^2 \approx 0.617\)
- \((484.99 - 483.562)^2 \approx 1.994\)
- \((484.85 - 483.562)^2 \approx 1.674\)
- \((439.49 - 483.562)^2 \approx 1917.823\)
Now summing these values:
\[
\sum (x_i - \mu)^2 \approx 2720.177
\]
So,
\[
\text{Variance} \approx \frac{2720.177}{15} \approx 181.345
\]
**Standard deviation:**
\[
\text{Standard deviation} = \sqrt{\text{Variance}} \approx \sqrt{181.345} \approx 13.451
\]
### Final Answers for July:
- Population standard deviation: **13.451**
- Population variance: **181.345**
Overall:
- June: Standard deviation = 12.688, Variance = 161.018
- July: Standard deviation = 13.451, Variance = 181.345