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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 devi...
Page348  Student’s t Distribution      The variable t is defined as follows:       Assume that x has a normal distributio...
Properties of a Student’s tDistribution1.   The distribution is symmetric about the mean 0.2.   The distribution depends o...
Page349  Finding Critical Values      Table 6 of Appendix II gives various t values for       different degrees of freedo...
Finding Critical Values: Using Table 61.   Find the column with the c heading2.   Compute the degrees of freedom and find ...
Page349  Example 4 – Student’s t Distribution       Find the critical value tc for a 0.99 confidence       level for a t d...
Page350  Confidence Interval for μ when σ is     Unknown    Requirements       Let x be a random variable appropriate to ...
Not in Textbook!   How To Construct a Confidence   Interval   1.   Check Requirements           Simple random sample?    ...
Page  Example 5 – Confidence351  Interval  
Solution – Confidence Interval         The archaeologist can be 99% confident that         the interval from 44.5 cm to 4...
Using the Calculator1.   Hit STAT, tab over TESTS, Choose 8:Tinterval2.   Highlight STATS, hit ENTER3.   Enter the request...
Page353       Summary: Which Distribution       Should You Use?
Assignment   Page 354   #1, 4 – 7, 11 – 15 odd
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7.2 estimate mu, sigma unknown

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7.2 estimate mu, sigma unknown

  1. 1. 7.2 ESTIMATING μWHEN σ IS UNKNOWN Chapter 7: Estimation
  2. 2. 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.
  3. 3. 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.
  4. 4. 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
  5. 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
  6. 6. 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.
  7. 7. 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
  8. 8. 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
  9. 9. 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
  10. 10. Page Example 5 – Confidence351 Interval 
  11. 11. 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.
  12. 12. 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)
  13. 13. Page353 Summary: Which Distribution Should You Use?
  14. 14. Assignment Page 354 #1, 4 – 7, 11 – 15 odd

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