Estimate the standard deviation of the Japanese car population (s or σ ̂). (Report the answer with 3 decimal points. Remember to use the formula for σ ̂  and not σ.  Provide your calculation process.)  σ ̂ = √[∑(X-M)2/(n-1)] when M=31.67 and N=15 and X=34,30,32,35,34,28,34,29,30,33,37,32,29,34,24

We are tasked with calculating the sample standard deviation \( \hat{\sigma} \) of the Japanese car population using the provided formula:

\[
\hat{\sigma} = \sqrt{\frac{\sum (X_i - M)^2}{n-1}}
\]

Where:
- \( M \) is the sample mean (given as \( M = 31.67 \)),
- \( n \) is the sample size ( \( n = 15 \)),
- \( X_i \) are the individual data points (the list of car counts),
- \( \sum (X_i - M)^2 \) represents the sum of the squared deviations from the mean.

### Step 1: Compute the squared deviations from the mean.
We calculate \( (X_i - M)^2 \) for each \( X_i \) in the list:

The list of \( X \) values: \( 34, 30, 32, 35, 34, 28, 34, 29, 30, 33, 37, 32, 29, 34, 24 \)

For each data point \( X_i \), calculate \( (X_i - 31.67)^2 \).

### Step 2: Sum the squared deviations.

### Step 3: Divide by \( n - 1 = 14 \).

### Step 4: Take the square root of the result to find the standard deviation.

Let me perform these calculations for you.The estimated standard deviation of the Japanese car population, \( \hat{\sigma} \), is approximately \( 3.331 \) (rounded to three decimal places).

### Calculation Process:
1. Squared deviations were calculated for each data point.
2. The sum of squared deviations was divided by \( n - 1 = 14 \).
3. The square root of this value was taken to find the standard deviation.

Would you like further details or have any questions? Here are some related questions:

1. What is the difference between \( \sigma \) and \( \hat{\sigma} \)?
2. How does increasing the sample size affect the standard deviation estimate?
3. Why do we divide by \( n-1 \) instead of \( n \) when estimating the sample standard deviation?
4. How do outliers impact the standard deviation?
5. Can the standard deviation be negative?

**Tip:** Always check whether you need the population standard deviation (\( \sigma \)) or the sample standard deviation (\( \hat{\sigma} \)) based on the context of your data.

Estimate the standard deviation of the Japanese car population (s or σ ̂). (Report the answer with 3 decimal points. Remember to use the formula for σ ̂ and not σ. Provide your calculation process.) σ ̂ = √[∑(X-M)²/(n-1)] when M=31.67 and N=15 and X=34,30,32,35,34,28,34,29,30,33,37,32,29,34,24

Learn how to estimate the sample standard deviation (σ ̂) for the Japanese car population using the formula σ ̂ = √[∑(X-M)²/(n-1)]. With a sample mean of 31.67, a sample size of 15, and specific car counts provided, follow the step-by-step calculation process to reach a standard deviation of 3.331. Ideal for students studying college-level statistics.

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