Statistics And Star Wars Exploring Student Movie Attendance At Mt San Jacinto College

Introduction: The Force Awakens Statistical Inquiry

Hey guys! Let's dive into a fascinating statistical scenario right here at Mt. San Jacinto College. Our statistics instructor has a hunch—a statistical hypothesis, if you will—that fewer than 13% of our fellow students braved the midnight crowds to catch the opening night showing of the latest Star Wars movie. To investigate this, she surveyed 120 students and discovered that 12 of them were indeed among the dedicated fans present at the midnight premiere. Now, the big question is: Does this sample data provide sufficient evidence to support her belief? This is where the magic of hypothesis testing comes into play, and we're going to break down the entire process, step by step, in a way that even the most statistically-challenged can follow. We’ll journey through defining hypotheses, calculating test statistics, determining p-values, and ultimately, drawing a conclusion that’s grounded in solid statistical reasoning. So, grab your lightsabers (or calculators!), and let's embark on this statistical adventure together!

Setting the Stage: Hypotheses and Significance Levels

To kick things off, it's crucial to frame our investigation with clear hypotheses. In statistical lingo, we have two main contenders: the null hypothesis and the alternative hypothesis. The null hypothesis, often denoted as H₀, is the status quo – the statement we're trying to disprove. In our Star Wars case, it's the idea that 13% or more of Mt. San Jacinto College students attended the midnight showing. Mathematically, we can express this as p ≥ 0.13, where 'p' represents the true proportion of students who went to the premiere. On the other side of the coin, we have the alternative hypothesis (H₁ or Ha), which is what the instructor suspects – that less than 13% of students attended. This is written as p < 0.13. This is a one-tailed test because we're specifically looking for evidence in one direction (less than 13%).

Now, before we crunch any numbers, we need to set our significance level, often denoted as α (alpha). This is the threshold we use to decide whether our results are statistically significant or just due to random chance. A common choice is α = 0.05, which means we're willing to accept a 5% risk of incorrectly rejecting the null hypothesis (a Type I error). Think of it as our margin of error for making a decision. If the evidence against the null hypothesis is strong enough (i.e., the probability of observing our results if the null hypothesis were true is less than 0.05), we'll reject it in favor of the alternative hypothesis. Choosing the significance level is a crucial step, as it dictates how much evidence we require to support our claim. A lower significance level (e.g., 0.01) makes it harder to reject the null hypothesis, while a higher level (e.g., 0.10) makes it easier. For our Star Wars saga, we'll stick with the standard α = 0.05, providing a balance between being cautious and detecting a real effect.

Diving into the Data: Calculations and Test Statistics

Alright, let's get our hands dirty with some actual calculations! Remember, our instructor surveyed 120 students, and 12 of them attended the midnight showing. This gives us a sample proportion (often denoted as p̂) of 12/120 = 0.10, or 10%. This is the proportion we observed in our sample, but the million-dollar question is: How likely is it to observe a sample proportion of 10% if the true proportion is actually 13% or higher (as stated in our null hypothesis)? To answer this, we need to calculate a test statistic. The test statistic will quantify how far our sample proportion deviates from the null hypothesis, taking into account the sample size and the variability in the data. For proportions, we typically use the z-test statistic, which follows a standard normal distribution under the null hypothesis.

The formula for the z-test statistic is:

z = (p̂ - p₀) / √((p₀(1 - p₀)) / n)

Where:

• p̂ is the sample proportion (0.10 in our case)
• p₀ is the hypothesized population proportion (0.13)
• n is the sample size (120)

Let's plug in the numbers:

z = (0.10 - 0.13) / √((0.13(1 - 0.13)) / 120)

z = -0.03 / √(0.1131 / 120)

z = -0.03 / √(0.0009425)

z = -0.03 / 0.0307

z ≈ -0.977

So, our z-test statistic is approximately -0.977. This value tells us how many standard deviations our sample proportion is away from the hypothesized proportion under the null hypothesis. A negative z-score indicates that our sample proportion is lower than the hypothesized proportion, which aligns with the instructor's suspicion. But, is this difference large enough to be statistically significant? That's where the p-value comes in.

Unveiling the P-Value: The Probability of the Observed

Now, let's decode the mystery of the p-value. The p-value is the probability of observing a sample proportion as extreme as (or more extreme than) ours, assuming the null hypothesis is true. In other words, it tells us how likely it is to see a sample proportion of 10% (or lower) if the true proportion of Star Wars fans is actually 13% or higher. A small p-value suggests that our observed result is unlikely under the null hypothesis, making us lean towards rejecting it.

Since we're conducting a left-tailed test (because our alternative hypothesis is p < 0.13), we need to find the area under the standard normal curve to the left of our z-test statistic (-0.977). This area represents the p-value. We can use a z-table, a statistical calculator, or software to find this probability. Using a z-table or calculator, we find that the p-value associated with a z-score of -0.977 is approximately 0.164.

This means that if the true proportion of students who attended the midnight showing was indeed 13% or higher, there's about a 16.4% chance of observing a sample proportion as low as 10% just due to random sampling variability. That's not a super small probability, which is a crucial clue for our final decision.

Drawing Conclusions: Reject or Fail to Reject?

Okay, guys, the moment of truth has arrived! We've crunched the numbers, calculated our test statistic, and unearthed the p-value. Now, we need to put it all together and make a decision about our instructor's hypothesis. This is where we compare our p-value (0.164) to our significance level (α = 0.05). Remember, the significance level is our threshold for statistical significance.

Here's the golden rule:

• If the p-value is less than or equal to the significance level (p ≤ α), we reject the null hypothesis. This means we have enough evidence to support the alternative hypothesis.
• If the p-value is greater than the significance level (p > α), we fail to reject the null hypothesis. This means we don't have enough evidence to support the alternative hypothesis.

In our Star Wars scenario, our p-value (0.164) is greater than our significance level (0.05). Therefore, we fail to reject the null hypothesis. This means that based on the survey data, we don't have enough statistical evidence to conclude that less than 13% of Mt. San Jacinto College students attended the midnight showing of the latest Star Wars movie.

It's important to emphasize that failing to reject the null hypothesis doesn't necessarily mean it's true. It simply means that our data doesn't provide strong enough evidence to reject it. There could be several reasons for this. Maybe the true proportion is indeed close to 13%, or perhaps our sample size of 120 students wasn't large enough to detect a smaller difference. Statistical conclusions are always made with a degree of uncertainty, and it's crucial to interpret them cautiously.

Beyond the Numbers: Interpreting the Results in Context

So, what does all this mean in plain English? Well, despite our instructor's initial hunch, the data from her survey doesn't give us a strong reason to believe that less than 13% of Mt. San Jacinto College students attended the Star Wars midnight premiere. It's possible that the actual percentage is around 13%, or even higher. Maybe Star Wars fever is alive and well at our college!

However, it's crucial to remember the limitations of our analysis. Our conclusion is based on a sample of 120 students, and it's always possible that this sample isn't perfectly representative of the entire student population. There are other factors that could influence attendance at a midnight movie showing, such as personal preferences, schedules, and access to transportation. To gain a more comprehensive understanding, we might consider conducting a larger survey, gathering data from a more diverse group of students, or even exploring other data sources, such as ticket sales or social media activity.

In the grand scheme of things, this statistical exercise highlights the power and limitations of hypothesis testing. It provides a structured framework for evaluating claims based on evidence, but it's just one piece of the puzzle. Context, critical thinking, and a healthy dose of skepticism are all essential ingredients for making informed decisions in the real world. And who knows, maybe next time, we'll gather even more data and revisit the Star Wars fandom at Mt. San Jacinto College with even greater statistical firepower!

Repair Input Keyword

Original Keyword: A statistics instructor believes that less than $13 \%$ of Mt. San Jacinto College students attended the opening night midnight showing of the latest Star Wars movie. She surveys 120 of her students and finds that 12 attended the midnight showing.

Rewritten Keyword: How can we statistically test if less than 13% of Mt. San Jacinto College students attended the Star Wars movie midnight showing, given that a survey of 120 students found 12 attendees?