A random sample of size twenty of a golfer’s scores are given:
39, 40, 40, 41, 39, 40, 44, 43, 40, 41, 40, 41,41,
42, 43, 40, 41, 41, 41, 43.
Compute the mean, sample variance and the sample standard deviation
for the sample of the golfer’s scores.

To compute the mean, sample variance, and sample standard deviation for the golfer’s scores, we'll use the following formulas:

1. **Mean (μ):**
   \[
   \mu = \frac{1}{n} \sum_{i=1}^{n} x_i
   \]
   where \(x_i\) are the individual scores and \(n\) is the sample size.

2. **Sample Variance (s^2):**
   \[
   s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \mu)^2
   \]
   where \(\mu\) is the mean of the sample.

3. **Sample Standard Deviation (s):**
   \[
   s = \sqrt{s^2}
   \]

Let's calculate these step by step.

Given scores:
\[ 39, 40, 40, 41, 39, 40, 44, 43, 40, 41, 40, 41, 41, 42, 43, 40, 41, 41, 41, 43 \]

The sample size \(n = 20\).

### Step 1: Calculate the Mean
\[
\mu = \frac{1}{20} (39 + 40 + 40 + 41 + 39 + 40 + 44 + 43 + 40 + 41 + 40 + 41 + 41 + 42 + 43 + 40 + 41 + 41 + 41 + 43)
\]

I'll calculate this now.The mean score (μ) is \( 41.0 \).

### Step 2: Calculate the Sample Variance (s²)
Next, we calculate the sample variance using the formula:
\[
s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \mu)^2
\]
I'll calculate this now.The sample variance (s²) is approximately \( 1.895 \).

### Step 3: Calculate the Sample Standard Deviation (s)
The sample standard deviation is the square root of the variance:
\[
s = \sqrt{s^2}
\]
I'll calculate this now.The sample standard deviation (s) is approximately \( 1.376 \).

### Summary of Results:
- **Mean (μ):** 41.0
- **Sample Variance (s²):** 1.895
- **Sample Standard Deviation (s):** 1.376

Would you like more details or have any questions?

Here are some related questions you might find interesting:
1. What is the interpretation of standard deviation in this context?
2. How does sample variance differ from population variance?
3. What impact does sample size have on variance and standard deviation?
4. How can outliers affect the mean, variance, and standard deviation?
5. What other statistical measures could be useful to analyze this data?
6. How would the results change if the sample size increased?
7. What is the difference between a sample and a population in statistics?
8. Can you explain the difference between descriptive and inferential statistics?

**Tip:** When analyzing sample data, it's crucial to distinguish between sample variance and population variance. Sample variance uses \(n-1\) as the denominator to account for the sample being a subset of the entire population.

Learn how to compute the mean, sample variance, and sample standard deviation of a golfer's scores with a step-by-step solution. Includes detailed calculations for the given sample data. Suitable for advanced high school students and anyone interested in statistics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation