7.2 ESTIMATING μWHEN σ IS UNKNOWN Chapter 7: Estimation
Page347 Usually, when is unknown, is unknown as well. In such cases, we use the sample standard deviation s to approximate . A Students t distribution is used to obtain information from samples of populations with unknown standard deviation. A Student’s t distribution depends on sample size.
Page348 Student’s t Distribution The variable t is defined as follows: Assume that x has a normal distribution with mean μ. For samples of size n with sample mean and sample standard deviation s, the t variable has a Student’s t distribution with degrees of freedom d.f. = n – 1 Each choice for d.f. gives a different t distribution.
Properties of a Student’s tDistribution1. The distribution is symmetric about the mean 0.2. The distribution depends on the degrees of freedom.3. The distribution is bell-shaped, but has thicker tails than the standard normal distribution.4. As the degrees of freedom increase, the t distribution approaches the standard normal distribution.5. The area under the entire curve is 1. Figure 7-5 A Standard Normal Distribution and Student’s t Distribution with d.f. = 3 and d.f. = 5
Page349 Finding Critical Values Table 6 of Appendix II gives various t values for different degrees of freedom d.f. We will use this table to find critical values tc for a c confidence level. In other words, we want to find tc such that an area equal to c under the t distribution for a given number of degrees of freedom falls between –tc and tc in the language of probability, we want to find tc such that P(–tc t tc) = c Figure 7-6 Area Under the t Curve Between –tc and tc
Finding Critical Values: Using Table 61. Find the column with the c heading2. Compute the degrees of freedom and find the row that contains the d.f.3. Match the column and row Convention for using Student’s t distribution If the d.f. you need are not in the table, use the closest d.f. in the table that is smaller.
Page349 Example 4 – Student’s t Distribution Find the critical value tc for a 0.99 confidence level for a t distribution with sample size n = 5. Student’s t Distribution Critical Values (Excerpt from Table 6, Appendix II) Table 7-3 t0.99 = 4.604
Page350 Confidence Interval for μ when σ is Unknown Requirements Let x be a random variable appropriate to your application. Obtain a simple random sample (of size n) of x values from which you compute the sample mean and the sample standard deviation s. If you can assume that x has a normal distribution or is mound- shaped, then any sample size n will work. If you cannot assume this, then use a sample size of n ≥ 30. Confidence Interval for μ when σ is unknown where = sample mean of a simple random sample d.f. = n – 1 = confidence level (0 < c < 1) = critical value
Not in Textbook! How To Construct a Confidence Interval 1. Check Requirements Simple random sample? Assumption of normality? Sample size? Sample mean? Sample standard deviation s? 2. Compute E 3. Construct the interval using
Solution – Confidence Interval The archaeologist can be 99% confident that the interval from 44.5 cm to 47.8 cm is an interval that contains the population mean for shoulder height of this species of miniature horse.
Using the Calculator1. Hit STAT, tab over TESTS, Choose 8:Tinterval2. Highlight STATS, hit ENTER3. Enter the requested information4. Highlight Calculate, Hit EnterNote: The solution will be listed in the format (lower value, upper value)
Page353 Summary: Which Distribution Should You Use?