Each possible random sample provides a possible

This preview shows 16 out of 23 pages.

Each possible random sample provides a possible sample mean value. ! If the population is normal or the sample size is large (n # 30), then the sample mean will have a distribution that is (approximately) normal with mean μ and standard deviation . ! About 95% of the possible sample mean values with be in the interval ! For each of these sample mean values, the interval will contain the true population mean μ . ! Thus, about 95% of the intervals will contain the true population mean μ , in repeated samples of the same size n from the same population. σ / n μ ± 1 . 96 σ n ¯ x ± 1 . 96 σ n ¯ x ± 1 . 96 σ n 16
Image of page 16

Subscribe to view the full document.

Applet #14 of the CD that comes with the textbook ! If our confidence level is 95%, then in the long run, 95% of our confidence intervals will contain ! and 5% will not. 17 Try this by yourselves.
Image of page 17
Note: about confidence intervals ! The population parameter is not a random quantity . It does not vary – once we have “looked” (computed) the actual interval, we cannot talk about probability or chance for the particular interval anymore. ! The 95% confidence level applies to the procedure , not to an individual interval; it applies “before you look” and not “after you look” at your data and compute your interval. 18
Image of page 18

Subscribe to view the full document.

100(1- ! )% confidence interval for μ For normal population with known σ : ¯ x ± z α / 2 ( σ n ) What if the population distribution is not normal but we know σ ? Based on CLT, if the sample size is large ( n 30) we can still construct the confidence interval by ¯ x ± z α / 2 ( σ n ) . What if σ is unknown? If the sample size n is large ( n 30), then s is a good estimate for σ . We will use ¯ x ± z α / 2 ( s n ) . What if σ is unknown, and the sample size is small? If we know the population is normal, we can use t-distribution to help us construct the confidence interval. 19
Image of page 19
Interpretation of a CI for μ 20 ! We are 100(1- ! )% confident that the population mean μ lies somewhere between the lower and the upper bounds of the confidence interval. ! In repeated sampling, 100(1- ! )% of the resulting intervals will contain the population mean μ .
Image of page 20

Subscribe to view the full document.

Commonly used Confidence Levels 21 1- " " " /2 Z " /2 0.90 0.10 0.05 1.645 0.95 0.05 0.025 1.96 0.98 0.02 0.01 2.33 0.99 0.01 0.005 2.575
Image of page 21
Try It! 22 ! Suppose that the amount of time teenagers spend working at part-time jobs is normally distributed with a standard deviation of 20 minutes. A random sample of 100 observations is drawn and the sample mean computed as 125 minutes. Determine the 95% confidence interval estimate of the population mean.
Image of page 22

Subscribe to view the full document.

Image of page 23
You've reached the end of this preview.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern