Question

Can you afford the price of an NBA ticket during the regular season? The website Www.answers.com indicates that the low prices are around \(\$ 10\) for the high up seats while the court-side seats are around \(\$ 2000\) to \(\$ 5000\) per game and the average price of a ticket is \(\$ 75.50\) a game. \(^{7}\) Suppose that we test this claim by selecting a random sample of \(n=50\) ticket purchases from a computer database and find that the average ticket price is \(\$ 82.50\) with a standard deviation of \(\$ 75.25 .\) a. Do you think that \(x\), the price of an individual regular season ticket, has a mound-shaped distribution? If not, what shape would you expect? b. If the distribution of the ticket prices is not normal, you can still use the standard normal distribution to construct a confidence interval for \(\mu,\) the average price of a ticket. Why? c. Construct a \(95 \%\) confidence interval for \(\mu,\) the average price of a ticket. Does this confidence interval cause you support or question the claimed average price of \(\$ 75.50 ?\) Explain.

Short answer

Why or why not?

Step-by-step solution

Distribution shape prediction

Since the tickets' prices range widely from \(10 to \)5000, we can expect that the distribution of the individual ticket prices may not be mound-shaped (normal distribution) as the prices for court-side seats are much higher than those for the high-up seats and average seats. It is likely that the distribution would be positively skewed, indicating that there would be more tickets with lower prices.

Explanation for using the standard normal distribution

Even if the distribution of the ticket prices is not normal, the Central Limit Theorem enables us to use the standard normal distribution to construct a confidence interval for the average ticket price. The Central Limit Theorem states that the distribution of the sample means will approach a normal distribution as the sample size n increases, regardless of the original distribution of the population. In this case, we have a sample size of 50, which is generally sufficient for the Central Limit Theorem to apply.

Calculate the margin of error

To construct a 95% confidence interval, we first need to calculate the margin of error. The formula for margin of error is: Margin of error = \(z^* * \frac{\text{standard deviation}}{\sqrt{\text{sample size}}}\) For a 95% confidence interval, the critical z-value (\(z^*\)) is approximately 1.96. Margin of error = \(1.96 * \frac{75.25}{\sqrt{50}}\)

Compute the margin of error

Now let's compute the margin of error: Margin of error = \(1.96 * \frac{75.25}{\sqrt{50}} \approx 20.61\)

Construct the confidence interval

With the margin of error calculated, we can now construct a 95% confidence interval for the average ticket price: Lower bound = Sample mean - Margin of error = \(82.50 - 20.61 \approx 61.89\) Upper bound = Sample mean + Margin of error = \(82.50 + 20.61 \approx 103.11\) So, the 95% confidence interval for the average ticket price is \((61.89, 103.11)\).

Comparing the confidence interval with the claimed average price

Finally, let's compare this confidence interval with the claimed average ticket price of \(75.50. Since the claimed average price falls within our 95% confidence interval, we have no reason to doubt the claimed average ticket price of \)75.50. This means that the claimed average price is plausible given our random sample of 50 ticket purchases.

Key concepts

Central Limit Theorem

Understanding the Central Limit Theorem (CLT) is crucial when working with sample data to make inferences about a larger population. Simply put, the CLT tells us that regardless of the shape of the underlying population distribution, the distribution of sample means will tend to be normal, or bell-shaped, as the sample size gets larger.

For practical purposes, statisticians agree that a sample size of 30 or more is often enough for the CLT to hold true. This is why, in our NBA ticket pricing exercise, with a sample size of 50, we can assume that the distribution of the sample mean will be approximately normal. This is a powerful concept because it allows us to use the techniques and formulas designed for the normal distribution to make statistical inferences about non-normally distributed data, as long as we're dealing with means from sufficiently large samples.

The CLT is the reason we can proceed with confidence interval construction even when the actual distribution of ticket prices is skewed, bearing in mind that the sample average will inhabit the properties of a normal distribution given a large enough sample.

Standard Normal Distribution

The standard normal distribution is a special case of the normal distribution where the mean is 0 and the standard deviation is 1. It's represented by the letter Z and is used widely in statistics because it has been tabulated extensively, thus providing a convenient way to calculate probabilities and percentiles.

In our context, we often need to convert our sample mean estimation into a standard normal form to use z-scores, which are a measure of how many standard deviations an element is from the mean. When constructing confidence intervals, we rely on these z-scores to find the critical values that determine our margin of error. For a 95% confidence interval, the z-score is approximately 1.96, which corresponds to the critical value for the standard normal distribution that bounds the central 95% of the distribution.

Sample Mean Estimation

Sample mean estimation is central to inferential statistics, as it allows us to use data from a small portion of the population to make estimates about the population average. In our example, the sample mean is the average ticket price obtained from the 50 recorded purchase prices. It serves as the best point estimate for the true average ticket price among all NBA game ticket prices.

However, because we're working with a random sample, there's always some level of uncertainty in our estimate. This is where the confidence interval comes in, providing a range within which we can say with a certain level of confidence that the population mean (the true average ticket price) lies. The accuracy of our sample mean estimate depends on the sample size and variability within the data, with larger samples and less variability typically leading to more accurate estimates.

Margin of Error Calculation

The margin of error is a statistic expressing the amount of random sampling error in a survey's results. It tells us how far we can expect our sample estimate to be from the true population value and is a vital part of constructing a confidence interval. Calculation of the margin of error incorporates the standard deviation of the sample and the sample size, as well as the critical value from the standard normal distribution (z-score) for the desired level of confidence.

The generic formula for the margin of error in our exercise is:
Margin of error = \(z^* \times \frac{{\text{{standard deviation}}}}{{\sqrt{{\text{{sample size}}}}}}\)
By inputting our known values—standard deviation, sample size, and the z-score for a 95% confidence interval—we can calculate the margin of error. In the NBA ticket price scenario, this calculation yielded a margin of error of approximately 20.61. This value tells us that we can be 95% confident that the true average ticket price lies within this range above and below our sample mean.