1. Find and interpret 95% confidence intervals for a mean when (A) the population is normal and (B) \(\sigma\) is known.

• A point estimate is a single-value estimate of a population parameter.

• We say that a statistic is an unbiased estimator if the mean of its distribution is equal to the population parameter.

  • Otherwise, it is a biased estimator.

• Ideally, we want estimates that are unbiased with small standard error.

  • For example, a sample mean (unbiased) with a large sample size (results in smaller standard error).

Point estimates are useful, but they only give us so much information. The variability of an estimate is also important!

Take a look at these two boxplots:

• Both samples are size \(n=100\) and have \(\bar{x}=0\)
• Variable 1 has a standard deviation of \(\sigma=0.5\)
• Variable 2 has standard deviation \(\sigma=5\)

A confidence interval is an interval of numbers based on the point estimate of the parameter (along with some other stuff).

• Say we want to be 95% confident about a statement.
• In Statistics, this means that we have arrived at our statement using a method that will give us a correct statement 95% of the time.

• Our best point estimate for \(\mu\) (based on a random sample) is \(\bar{x}\), so that value will make up the center of the interval.
• To create an interval around \(\bar{x}\), we will construct what is called the margin of error.
  • We will use the variability of the data along with some normal distribution properties.
  • This will look like \[z\times\frac{\sigma}{\sqrt{n}}\]
  • The value \(z\) will come from the normal distribution and will be based on how confident we want to be, e.g., 95% confident.

Putting everything together, the 95% confidence interval is \[\left(\bar{x} - z_*\frac{\sigma}{\sqrt{n}}, \bar{x} + z_*\frac{\sigma}{\sqrt{n}}\right)\] where \(z_* = 1.96\).

The value \(1.96\) is chosen because \((-1.96 < Z < 1.96) = 0.95\) (this is what makes it a 95% confidence interval!).

If an experiment is run infinitely many times, the true value of \(\mu\) will be contained in 95% of the intervals.

• It is NOT accurate to say that “the probability that \(\mu\) is in the confidence interval is 0.95”.
  • The parameter \(\mu\) is some fixed quantity and it’s either in the interval or it isn’t.
• We are NOT “95% confident that \(\bar{x}\) is in the interval”.
  • The value \(\bar{x}\) is some known quantity and it’s always in the interval.

The preferred keyboard height for typists is approximately normally distributed with \(\sigma=2.0\). A sample of size \(n=31\), resulted in a mean preferred keyboard height of \(80 cm\).

1. Find and interpret a 95% confidence interval for keyboard height.
2. What is the width of the interval? If we wanted a narrower interval, what could we do?

Suppose I took a random sample of 50 Sac State students and asked about their SAT scores and found a mean score of 1112. Prior experience with SAT scores in the CSU system suggests that SAT scores are well-approximated by a normal distribution with standard deviation known to be 50.

1. Find a 95% confidence interval for Sac State SAT scores.
2. Interpret your interval in the context of the problem.

Section 6.2 Exercises 1 and 2