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A histogram is a type of bar graph that represents the frequency distribution of a set of data. It gives a visual impression of the data's distribution by showing the number of data points that fall within a particular range of values, known as bins. To create a histogram for the drive-in movie data:

• Divide the sorted data into several bins. For instance, if you choose 5-6 bins, each will cover a specific range of the data.
• Count how many data points fall into each bin.
• Plot the bins on the x-axis and the frequency of data points on the y-axis.

When you observe the plotted histogram, look for a pattern that resembles a bell curve. A bell-shaped histogram indicates that the data may be normally distributed. However, the histogram alone cannot definitively determine normality.

The mean is a measure of central tendency, often referred to as the average. It provides a central value for a data set, to which individual data points tend to cluster. For the given drive-in movie data, the mean is calculated by summing up all the values and then dividing by the total number of observations. This formula is:\[\text{Mean} = \frac{\text{Sum of all data points}}{\text{Number of data points}}\]For our data, the sum of all values is 14,038, and there are 14 data points, resulting in a mean of 1002.71. Calculating the mean is essential because it serves as a reference point when analyzing how data is spread around it. This is especially important in the context of normality testing.

The standard deviation is a measure that describes the amount of variation or dispersion in a set of data. It tells us how much the individual data points deviate from the mean. A small standard deviation means that data points are close to the mean, while a large one indicates a wide spread.To calculate standard deviation:

• Subtract the mean from each data point to get the deviation score.
• Square each deviation score.
• Sum those squared deviations.
• Divide by the number of observations minus 1 (n-1 for a sample).
• Finally, take the square root of this value.

The formula is: \[\text{Standard Deviation} = \sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2}\] For the movie data, this results in approximately 382.8. This value helps quantify the expected deviation of a randomly selected data point from the mean, which is crucial for gauging normality.

A normal probability plot, often called a Q-Q plot, is a graphical tool to help assess if a data set is approximately normally distributed. In a Q-Q plot, each data point is plotted against the corresponding quantile value of a theoretical normal distribution. To create a Q-Q plot for the drive-in movie data:

• Plot each data point's percentile rank within the data set.
• On the x-axis, represent theoretical percentiles from a standard normal distribution.
• On the y-axis, represent the actual data percentiles.
• If the data is normally distributed, the points should follow roughly a straight diagonal line.

Deviations from the line indicate departures from normality. The Q-Q plot is highly informative as it allows for visual judgment, but it is usually used in conjunction with statistical tests for robust conclusions.

The Shapiro-Wilk test is a statistical test that assesses the normality of a data distribution. It's a well-regarded test due to its robustness and suitability for small sample sizes. To perform the Shapiro-Wilk test:

• Formulate the null hypothesis as "the data is normally distributed."
• Calculate the test statistic based on the data's ordered values (from smallest to largest).
• Compare the test statistic to a standard distribution of the statistic to find the p-value.
• Evaluate the p-value against a chosen significance level, typically 0.05.

If the p-value is greater than 0.05, we fail to reject the null hypothesis, suggesting the data does not significantly deviate from normality. For the drive-in movie data, if the calculated p-value exceeds 0.05, it would indicate a normal distribution, helping to validate your histogram and Q-Q plot findings.