Section 9.3: Confidence Interval for a Population Mean

One Sample z Confidence Interval for μ The general formula for a confidence interval for a population mean μ is: When, 1.X bar is the sample mean from a random sample 2.The sample size n is large (generally n ≥ 30) 3.σ, the population standard deviation, is known

Important Properties of t Distributions 1.The t curve corresponding to any fixed number of degrees of freedom is bell shaped and is centered at 0 (just like the standard normal (z) curve). 2.Each t curve is more spread out than the z curve. 3.As the number of degrees of freedom increases, the spread of the corresponding t curve decreases. 4.As the number of degrees of freedom increases, the corresponding sequence of t curves approaches the z curve

Let x1, x2, …, xn constitute a random sample from a normal population distribution. Then the probability distribution of the standardized variable

One Sample t Confidence Interval for μ The general formula for a confidence interval for a population mean μ based on a sample of size n is When 1.X bar is the sample mean from a random sample 2.The population distribution is normal, or the sample size n is large 3.σ, the population standard deviation is unknown

Example A study of the ability of individuals to walk in a straight line reported the following data on cadence for a sample of n = 20 randomly selected healthy men:

This is a normal probability plot since the plot is reasonably straight. So the calculates required are x bar and s

The t critical value for a 99% confidence interval based on 19 df is The interval is:

The sample size required to estimate a population mean μ to within an amount B with 95% confidence is: If σ is unknown, it can be estimated based on previous information or, for a population that is not too skewed, by using (range)/4.

Example The financial aid office wishes to estimate the mean cost of textbooks per quarter for students at a particular university. For the estimate to be useful, it should be within $20 of the true population mean. How large a sample should be used to be 95% confident of achieving this level of accuracy?

To determine the required sample size we must have a value for σ. The financial aid office is pretty sure that the amount spent on books varies widely, with most values between $50 and $450. We will use the range to find a reasonable sigma value.

We can now use 100 as the σ Rounding up is always necessary. So we would need a sample size of 97 or larger.