The population mean is 102.4. The population standard deviation is 106.5. What percentage of the 50 regions have stores within plus minus 1 , 2, and 3 standard deviations of the mean?

To determine the percentage of regions with stores within \(\pm 1\), \(\pm 2\), and \(\pm 3\) standard deviations of the population mean, we'll follow these steps:

### Step 1: Calculate the boundaries
1. **Within 1 standard deviation:**
   - Lower bound: \( \text{Mean} - 1 \times \text{Standard Deviation} = 102.4 - 106.5 = -4.1 \)
   - Upper bound: \( \text{Mean} + 1 \times \text{Standard Deviation} = 102.4 + 106.5 = 208.9 \)

2. **Within 2 standard deviations:**
   - Lower bound: \( \text{Mean} - 2 \times \text{Standard Deviation} = 102.4 - 2(106.5) = -110.6 \)
   - Upper bound: \( \text{Mean} + 2 \times \text{Standard Deviation} = 102.4 + 2(106.5) = 315.4 \)

3. **Within 3 standard deviations:**
   - Lower bound: \( \text{Mean} - 3 \times \text{Standard Deviation} = 102.4 - 3(106.5) = -217.1 \)
   - Upper bound: \( \text{Mean} + 3 \times \text{Standard Deviation} = 102.4 + 3(106.5) = 421.9 \)

### Step 2: Count the number of data points within each range
Now, we'll check how many of the given data points fall within these ranges.

Given dataset: 
\[
67, 22, 127, 99, 159, 68, 20, 117, 40, 23, 75, 29, 160, 344, 90, 51, 198, 80, 136, 408, 561, 126, 49, 8, 42, 82, 69, 98, 201, 16
\]

### 1. **Within 1 standard deviation (-4.1 to 208.9):**

The values within this range are: 
67, 22, 127, 99, 159, 68, 20, 117, 40, 23, 75, 29, 160, 90, 51, 198, 80, 136, 126, 49, 8, 42, 82, 69, 98, 201, 16

Number of values: 27

Percentage: \(\frac{27}{30} \times 100\% = 90\%\)

### 2. **Within 2 standard deviations (-110.6 to 315.4):**

The values within this range are:
All the previous values plus 344.

Number of values: 28

Percentage: \(\frac{28}{30} \times 100\% = 93.33\%\)

### 3. **Within 3 standard deviations (-217.1 to 421.9):**

The values within this range are:
All the previous values plus 408.

Number of values: 29

Percentage: \(\frac{29}{30} \times 100\% = 96.67\%\)

### Summary:
- **Within 1 standard deviation:** 90%
- **Within 2 standard deviations:** 93.33%
- **Within 3 standard deviations:** 96.67%

Would you like more details or have any further questions?

Here are eight related questions:
1. Would you like to calculate the percentage within \(\pm 4\) standard deviations?
2. Should we plot a histogram of the dataset to visualize the distribution?
3. Are you interested in finding the skewness or kurtosis of this data?
4. Do you want to check how closely this data follows a normal distribution?
5. Shall we calculate the z-scores for each data point?
6. Would you like to see the cumulative distribution function (CDF) for this data?
7. Are you interested in performing a hypothesis test on this data?
8. Should we calculate the coefficient of variation to understand relative variability?

**Tip:** When working with normally distributed data, about 68%, 95%, and 99.7% of the data falls within 1, 2, and 3 standard deviations from the mean, respectively (empirical rule).

Calculate the percentage of regions within plus minus 1, 2, and 3 standard deviations of a population mean using statistical concepts. Understand the normal distribution and apply the empirical rule to analyze data.

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