Review Test for Chapters 8 and 9

Hello Math Class! This post provides a review for your upcoming test covering Chapters 8 and 9. Make sure to review the class notes and work through the practice problems to solidify your understanding of the material. Remember, practice makes perfect!

Key Concepts

  • Confidence Intervals: A range of values that, with a certain level of confidence, contains the true population parameter. We often use this to estimate the population mean, $\mu$.
  • Hypothesis Testing: A method for testing a claim about a population parameter. This involves setting up a null hypothesis ($H_0$) and an alternative hypothesis ($H_a$), calculating a test statistic, and determining whether to reject the null hypothesis based on a chosen significance level, $\alpha$.
  • One-Mean t-Test: Used to perform a hypothesis test for a population mean, $\mu$, when the population standard deviation, $\sigma$, is unknown. The test statistic is calculated as: $$t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$$, where $\bar{x}$ is the sample mean, $\mu_0$ is the hypothesized population mean, $s$ is the sample standard deviation, and $n$ is the sample size. The degrees of freedom (df) are $n-1$.

Practice Problems (Examples from Class Notes)

Here are some example problems similar to what you might see on the test. Remember to show your work!

  1. Political Prisoners:

    A study on political prisoners with chronic PTSD found that the mean duration of imprisonment for 32 patients was 33.4 months. Assuming $\sigma = 42$ months, determine a 95% confidence interval for the mean duration of imprisonment, $\mu$, of all East German political prisoners with chronic PTSD. Interpret your answer in words.

  2. Bottlenose Dolphins:

    A random sample of 50 adult bottlenose dolphins have a mean length of 12.04 ft with a standard deviation of 1.03 ft. Find and interpret a 90% confidence interval for the mean length of all adult bottlenose dolphins.

  3. Early-Onset Dementia:

    A simple random sample of 21 people with early-onset dementia gave a sample mean age at diagnosis of 52.5 years. Assume the population standard deviation is 6.8 years. At the 1% significance level, do the data provide sufficient evidence to conclude that the mean age at diagnosis of all people with early-onset dementia is less than 55 years old?

  4. Dirt Bikes:

    A random sample of 30 dirt bikes have a mean fuel capacity of 1.91 gallons with a standard deviation of 0.74 gallons. At the 10% significance level, do the data provide sufficient evidence to conclude that the mean fuel tank capacity of all dirt bikes is less than 2 gallons?

  5. Death Rolls:

    A sample of 20 alligator death rolls yielded a sample mean angle between the body and head of 49.0 degrees with a sample standard deviation of 10.0 degrees. At the 5% significance level, do the data provide sufficient evidence to conclude that, on average, the angle between the body and head of an alligator during a death roll is greater than 45°?

  6. Acid Rain and Lake Acidity:

    pH levels were measured for 15 high mountain lakes. The sample mean was found to be 6.6 with a sample standard deviation of 0.67. At the 5% significance level, do the data provide sufficient evidence to conclude that, on average, high mountain lakes in the Southern Alps are nonacidic (pH > 6)?

  7. TV Viewing:

    A random sample of 20 people watched an average of 4.760 hours of television per day last year with a standard deviation of 2.297 hours. In 2005, the average was 4.55 hours. At the 10% significance level, do the data provide sufficient evidence to conclude that the amount of television watched per day last year by the average person differed from that in 2005?

  8. Golf Robots:

    A golf club was tested using a robot to hit a ball six times. The total yards each ball traveled were as follows: 180, 187, 181, 182, 185, 181. The sample mean was 182.7 yards and the sample standard deviation was 2.73 yards. At the 5% significance level, do the data provide sufficient evidence to conclude that the club does what the golfer wants (hits the ball more than 180 yards on average)?

  9. Nursing-Home Costs:

    A random sample of 11 nursing homes yielded a sample mean daily cost of $267.6 with a sample standard deviation of $59.05 for a private room. In 2011, the average cost was $239. At the 5% significance level, does this year's average cost for a private room in a nursing home exceeds that in 2011?

Good luck with your test! Remember to review your notes, practice these problems, and come prepared to show what you've learned! You got this!