To determine the distribution shape of the given data set using a normal probability plot, histogram, and by comparing the mean and median, we can follow these steps:

Data Set:

$\{4, 5, 5, 2, 5, 5, 5, 4, 4, 6, 6, 7, 8, 7\}$

1. Normal Probability Plot

A normal probability plot helps us assess whether the data follows a normal distribution. If the data points in the plot form a roughly straight line, the data is approximately normally distributed. Deviations from a straight line suggest skewness or other departures from normality.

Steps to construct:

1. Rank the data in ascending order.
2. Calculate the cumulative probabilities for each rank.
3. Match the cumulative probabilities with the corresponding z-scores (standard normal quantiles).
4. Plot the data values against the z-scores.

I can compute the required z-scores for you, but let's move on to the histogram for more insight.

2. Histogram

A histogram allows us to visually assess the distribution of the data.

Binned Data:

• $2$ occurs once.
• $4$ occurs three times.
• $5$ occurs five times.
• $6$ occurs twice.
• $7$ occurs twice.
• $8$ occurs once.

The histogram would show a concentration of values around 5, with fewer values on the tails, possibly indicating some skewness.

3. Comparison of Mean and Median

• Mean: The average of the data set.
• Median: The middle value when the data set is ordered.

Let’s calculate these: $\text{Mean} = \frac{2 + 4 + 4 + 4 + 5 + 5 + 5 + 5 + 5 + 6 + 6 + 7 + 7 + 8}{14}$ $\text{Median} = \frac{5 + 5}{2} = 5$

Given that the mean is slightly higher than the median, this indicates a potential right skew (since the mean is pulled toward higher values).

Conclusion:

By using the normal probability plot, histogram, and comparing the mean and median, this data set appears to follow a distribution that is skewed right.

Would you like me to go into further detail on how to construct the normal probability plot, or clarify any of the steps? Here are some questions to deepen understanding:

1. How do you calculate the z-scores for a normal probability plot?
2. What is the impact of skewness on statistical analysis?
3. Can a distribution be normal if its mean and median differ?
4. How does the histogram reveal skewness in the data?
5. Why is it important to assess normality in data analysis?

Tip: A normal probability plot is often more reliable for small data sets when determining normality than relying solely on histograms or mean-median comparisons.