Chapter 8 and 9 Review
Welcome to the review for Chapters 8 and 9! Let's solidify your understanding of confidence intervals and hypothesis testing with some examples. Remember, practice is key to mastering these concepts, so let's dive in!
Key Concepts
- Confidence Intervals: Estimating population parameters (like the mean, $\mu$) using sample data. The general formula is: Sample Statistic $\pm$ (Critical Value)(Standard Error)
- Hypothesis Testing: Evaluating evidence to support or reject a claim (null hypothesis, $H_0$) about a population. This involves calculating a test statistic (like $z$ or $t$) and comparing it to a critical value or p-value.
Example Problems
1. Confidence Interval - Political Prisoners
A study examined the mean duration of imprisonment for 32 East German political prisoners with chronic PTSD, finding a sample mean of 33.4 months. Assuming the population standard deviation ($\sigma$) is 42 months, let's determine a 95% confidence interval for the population mean ($\mu$).
The 95% confidence interval is calculated to be: $33.4 \pm (1.96)(\frac{42}{\sqrt{32}}) = 33.4 \pm 14.61 $
This yields an interval of approximately 18.79 to 48.01 months. We are 95% confident that the true mean duration of imprisonment for all East German political prisoners with chronic PTSD falls within this range.
2. Confidence Interval - Bottlenose Dolphins
A random sample of 50 adult bottlenose dolphins had a mean length of 12.04 ft with a standard deviation of 1.03 ft. Let's find and interpret a 90% confidence interval for the mean length of all adult bottlenose dolphins.
Here, we use the t-distribution because the population standard deviation is unknown. The formula is $\bar{x} \pm t_{\alpha/2}(\frac{s}{\sqrt{n}})$ where $\bar{x} = 12.04$, $s = 1.03$, $n = 50$
Degrees of freedom $df = n-1 = 49$. The 90% confidence interval is approximately 11.79 ft to 12.29 ft.
3. Hypothesis Testing - Early-Onset Dementia
We want to test if the mean age at diagnosis of early-onset dementia is less than 55 years. A sample of 21 people had a mean age of 52.5 years, with a population standard deviation of 6.8 years. We'll use a 1% significance level.
Null Hypothesis ($H_0$): $\mu = 55$
Alternative Hypothesis ($H_a$): $\mu < 55$
The test statistic is calculated as: $z = \frac{52.5 - 55}{\frac{6.8}{\sqrt{21}}} = -1.68$
Since the critical value for a one-tailed test at $\alpha = 0.01$ is -2.33 and our test statistic (-1.68) is greater than -2.33, we fail to reject the null hypothesis. There isn't sufficient evidence to conclude that the mean age at diagnosis is less than 55 years.
4. Hypothesis Testing - Dirt Bikes
Is the mean fuel tank capacity of all dirt bikes less than 2 gallons? A random sample of 30 dirt bikes has a mean of 1.91 gallons with a standard deviation of 0.74 gallons. Use a 10% significance level.
Null Hypothesis ($H_0$): $\mu = 2$
Alternative Hypothesis ($H_a$): $\mu < 2$
The test statistic is: $t = \frac{1.91 - 2}{\frac{0.74}{\sqrt{30}}} = -0.67$
The critical t-value for a one-tailed test with 29 degrees of freedom at a 10% significance level is approximately -1.311. Because -0.67 is greater than -1.311, we fail to reject the null hypothesis. The data does not provide enough evidence to conclude the mean fuel tank capacity of all dirt bikes is less than 2 gallons.
5. Hypothesis Testing - Alligator Death Rolls
Is the average angle between the body and head of an alligator during a death roll greater than 45°? A sample of 20 rolls yielded a mean of 49° and a standard deviation of 10°. Use a 5% significance level.
Null Hypothesis ($H_0$): $\mu = 45$
Alternative Hypothesis ($H_a$): $\mu > 45$
The test statistic is: $t = \frac{49 - 45}{\frac{10}{\sqrt{20}}} = 1.79$
The critical t-value for a one-tailed test with 19 degrees of freedom at a 5% significance level is approximately 1.729. Because 1.79 is greater than 1.729, we reject the null hypothesis. There *is* sufficient evidence to conclude that the average angle is greater than 45°.
Keep practicing, and you'll be well-prepared for your test. Good luck, Professor Baker's Math Class!