91Ó°ÊÓ

When we talk about a confidence interval, we're discussing a range in which we believe the true mean of a population lies, based on our sample data. It’s like saying, "We’re fairly sure that the true answer is somewhere around here."
For our exercise, we constructed a 95% confidence interval for the average number of years students in the city take piano lessons. This interval was calculated to be \[ (3.57, 5.63) \]. This means we are 95% confident the true average is between 3.57 and 5.63 years.
The process involves using the sample mean, sample size, and standard deviation to find this range. Confidence intervals are crucial because they give us a way to quantify uncertainty.

• 95% here means if we were to take many samples and build an interval each time, about 95% of those intervals would contain the true mean.
• The wider the interval, the less precise our estimate. Conversely, a narrow interval means a more precise estimate.

In conclusion, the confidence interval in this exercise shows that while there’s a possibility the average years of lessons is below 5, it’s not definitive.

The t-test is a statistical test used to determine whether there is a significant difference between the sample mean and a known value, often the population mean. In our example, we're using a one-sample t-test to evaluate if the average years of piano lessons is less than 5.
To perform a t-test, the following steps are included:- **Calculating the t-statistic**: Formula used is \[ t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}} \], where \( \bar{x} \) is the sample mean, \( \mu_0 \) is the hypothesized population mean (5 years in this case), \( s \) is the sample standard deviation, and \( n \) is the sample size.
- **Finding the critical value**: Using a t-distribution table, matched against the chosen significance level.
For Georgianna's claim, the calculated t-value was about \[ -0.813 \]. With 19 degrees of freedom, our critical t-value at \( \alpha = 0.05 \) was \( -1.729 \).
The t-test helps determine if the sample provides enough evidence to reject the null hypothesis. In our case, the fact that the calculated t-value wasn’t extreme enough means Georgianna’s claim couldn’t be strongly supported.

The null hypothesis (\( H_0 \)) is a statement we seek to test. It often represents a status quo or default position and is assumed true until evidence suggests otherwise. In our piano lesson scenario:\[ H_0: \mu \geq 5 \]- Meaning, the average duration of piano lessons is 5 years or more.
The alternative hypothesis (\( H_a \)) opposes this:\[ H_a: \mu < 5 \]- Indicating the average is less than 5 years.
The null hypothesis serves as a benchmark against which the evidence is measured. It's crucial because rejecting it requires solid statistical evidence. In our test, since we failed to reject the null, it suggests no strong evidence exists to claim the average is less than 5.- **Why retain the null?**: Failure to find strong evidence against it, not proof it's true.
Null hypotheses often emphasize looking at data objectively, without bias. It ensures decisions are based on solid evidence, minimizing risks of false claims.

Significance level, denoted by \( \alpha \), is a threshold set before conducting hypothesis testing to decide whether an observed effect is genuine or likely due to chance. It’s often set at 0.05, as in our exercise.- **Purpose**: Helps control the probability of making a Type I error, which happens when the null hypothesis is wrongly rejected.

The alpha level defines how extreme the test statistic must be to consider evidence sufficient for rejecting \( H_0 \). In our study, we adhered to:- **\( \alpha = 0.05 \)**: Enables us to accept a 5% chance of committing a Type I error.
This threshold ensures the test's reliability. By comparing the \( t \)-statistic to the critical value from the t-distribution, a decision is made.

• Calculated t-value exceeding critical value leads to rejecting the null.
• Here, where \( -0.813 > -1.729 \), the results don't justify rejecting \( H_0 \).

Significance levels are crucial to balance between discovering true effects and reducing false alarms.