Question

What Is an Average Budget for a Hollywood Movie? Data 2.7 on page 95 introduces the dataset HollywoodMovies, which contains information on more than 900 movies that came out of Hollywood between 2007 and \(2013 .\) We will consider this the population of all movies produced in Hollywood during this time period. (a) Find the mean and standard deviation for the budgets (in millions of dollars) of all Hollywood movies between 2007 and \(2013 .\) Use the correct notation with your answer. (b) Use StatKey or other technology to generate a sampling distribution for the sample mean of budgets of Hollywood movies during this period using a sample size of \(n=20\). Give the shape and center of the sampling distribution and give the standard error.

Short answer

The mean and standard deviation give a numerical description of the dataset's central tendency and spread, respectively; while the sampling distribution, which can be generated and described using StatKey or similar technology, provides insight into the likelihood of different budget means for sampled groups of \(n=20\) movies.

Step-by-step solution

Find the Mean and Standard Deviation

To find the mean, sum up all the budgets within the dataset and then divide by the number of movies (data points). For the standard deviation, first find the variance by getting each budget's deviation from the mean, squaring it and finding the average of those squared deviations. The standard deviation is the squareroot of the variance.

Generate a Sampling Distribution

To generate a sampling distribution, use the given sample size (\(n=20\)) to form multiple samples from your population. Calculate the mean of each of these samples (the sample means) and plot them. This plot is your sampling distribution.

Describe the Sampling Distribution

The central tendency of a distribution is described by its mean (computed as the average of sample means). The spread is generally measured by the standard deviation, which in this case is the standard error. The standard error is the standard deviation of the sampling distribution. The shape of the distribution can be described as roughly smooth and symmetrical, bell-shaped (if it follows the normal distribution), skewed to the right or the left, or having outliers, depending on the resulting appearance of the plot.

Key concepts

Mean and Standard Deviation

When we speak of the average budget of Hollywood movies, we're referring to the mean, which is the sum of all movie budgets divided by the number of movies. This gives us a single value representing the typical amount spent on a movie production in Hollywood during the time studied, from 2007 to 2013.

The standard deviation is a powerful statistic that tells us how spread out the movie budgets are around this mean. To find it, as explained in the textbook solution, each movie's budget is compared to the mean by calculating the budget's deviation from the mean, then this deviation is squared. We do this for all movie budgets to mitigate the effect of differences being above or below the mean. Finally, after averaging these squared deviations, we get the variance and take its square root to return to our original unit of measurement, obtaining the standard deviation.

This gives us an insight into the variability of Hollywood movie budgets. A larger standard deviation indicates that the budgets are more spread out from the mean, showing a greater diversity in how much movies might cost to produce.

Sampling Distribution

If we were to randomly choose samples of 20 movies from the Hollywood dataset and calculate the mean budget for each sample, the collection of these mean values would form what statisticians call a sampling distribution. This distribution provides a visual representation of the means of all possible samples of a given size from our population.

The shape of this sampling distribution gives us important information. If the original population of movie budgets is normally distributed, and the sample size is sufficiently large, the Central Limit Theorem suggests that our sampling distribution of the mean will also be normally distributed — that is, bell-shaped and symmetrical. However, if the population distribution is not normal or the sample size is small, the sampling distribution may take on a different shape, such as being skewed or having outliers.

The central tendency of a sampling distribution is often described by its own mean, which should align closely with the population mean if the sample size is adequate and the samples are properly randomized.

Standard Error

The standard error is a crucial measure to understand in the context of sampling distributions. It tells us how much the sample means vary from the actual population mean. Essentially, it's the standard deviation for the sampling distribution of the mean, not to be confused with the standard deviation of the original data.

Why do we care about the standard error? It gives us a gauge of precision for sample estimates of the population mean. A smaller standard error implies that the sample mean is a more accurate reflection of the population mean. When conducting a study like the one on Hollywood movie budgets, knowing the standard error helps researchers understand the reliability of their estimates based on samples.

In the context of our Hollywood movie budgets, the standard error is calculated using the sample means of budgets from samples of 20 movies each. If you pick a different sample of 20 movies, chances are you will get a slightly different sample mean, and the standard error quantifies just how much 'slightly different' we can expect on average.