Let's break down the steps and compute everything using the provided data set.

1. Data Set

6, 15, 4, 5, 4, 31, 6, 5, 24, 8, 8, 44, 10, 10, 39, 0, 27, 5, 9, 5, 21, 7, 21, 9, 28, 16, 52, 3, 10, 30, 0, 3, 6, 12, 4, 7, 7, 14, 8, 15

2. Finding Statistics

Using this data set, I'll compute the following:

Sample Size ($n$):

The sample size is simply the total number of data points in the set.

Mean ($\mu$):

The mean is calculated by summing all the values in the dataset and dividing by the number of data points.

$\mu = \frac{\sum{x_i}}{n}$

Median:

The median is the middle value when the data points are sorted in ascending order. If there is an even number of data points, the median is the average of the two middle numbers.

Mode:

The mode is the value that appears most frequently in the data set.

Standard Deviation ($\sigma$):

The standard deviation measures the spread of the data points around the mean.

$\sigma = \sqrt{\frac{\sum{(x_i - \mu)^2}}{n}}$

Variance ($\sigma^2$):

Variance is the square of the standard deviation.

$\sigma^2 = \frac{\sum{(x_i - \mu)^2}}{n}$

Range:

The range is the difference between the largest and smallest data points.

$\text{Range} = \text{Max} - \text{Min}$

Let me compute these statistics for your data set.### Statistical Results:

1. Sample Size (n): 40
2. Mean: 13.45
3. Median: 8.5
4. Mode: 5
5. Standard Deviation: 12.12
6. Variance: 146.95
7. Range: 52 (from 52 - 0)

Now, let's create the histogram for this data set and describe its shape.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, based on the statistics we computed earlier, you can create the histogram using software like StatCrunch, Excel, or other statistical tools by using the data set.

Histogram Shape:

You can expect the shape of the histogram to be right-skewed (positively skewed) because:

• The mean (13.45) is larger than the median (8.5).
• There are larger outliers (like 44, 52) pulling the mean up.

Prediction of Shape from Statistics:

Yes, the skewness could have been predicted by comparing the mean and median. When the mean is greater than the median, it typically indicates a right-skewed distribution due to the influence of larger values.

Useful Measure of Center:

The median is likely the more useful measure of center for this data set because the data is skewed. The median is less sensitive to extreme values (outliers), while the mean can be affected by them, making the median a better representation of the typical wait time.

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Here are five questions related to the analysis:

1. How do outliers affect the mean and median in this data set?
2. What would happen if the highest values (like 52 and 44) were removed from the data?
3. How does the skewness of a distribution impact the choice of center (mean vs. median)?
4. Can the mode be a useful measure of center in skewed distributions?
5. How does the standard deviation help in understanding the spread of data?

Tip: In right-skewed distributions, consider using the median over the mean as it reduces the impact of extreme values on your analysis.