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Section 8.2 Estimating When is Unknown In order to use the normal distribution to find the confidence intervals for a population mean μ, we need to x know the value of σ, the population standard deviation. However much of the time, when μ is unknown, σ is unknown as well. In such cases, we use the sample standard deviation s to approximate σ. When we use s to approximate σ, the sampling distribution for x follows a new distribution called a Student’s t distribution. 8.2 / 1 Student’s t Distributions The student’s t distribution uses a variable t defined as follows. A student’s t distribution depends on sample size n. Assume that x has a normal distribution with mean μ. For samples of size n with sample mean x and standard deviation s, the t variable x has a Student’s t distribution with t s degree of freedom d.f. = n – 1. n 8.2 / 2 The shape of the t distribution depends only on the sample size, n, if the basic variable x has a normal distribution. When using the t distribution, we will assume that the x distribution is normal. Appendix Table 4 (Page A8) gives values of the variable t corresponding to the number of degrees of freedom (d.f.) d.f. = n – 1 where n = sample size 8.2 / 3 The t Distribution has a Shape Similar to that of the the Normal Distribution 4 Properties of a Student’s t Distribution 1.The distribution is Symmetric about the mean 0. 2.The distribution depends on the degrees of freedom (d.f. = b-1 for μ confidence intervals. 3.The distribution is Bell-shaped with thicker tails than the standard normal distribution. 4.As the degrees of freedom increase, the t distribution approaches the standard normal distribution 8.2 / 5 Using Table 4 to find Critical Values for confidence Intervals Table 4 of the Appendix gives various t values for different degrees of freedom d.f. We will use the table to find the critical values tc for a c confidence level. In other words, we want to find tc such thatt the 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 c P(tc t tc ) c This probability corresponds to the shaded area in next figure 8.2 / 6 Using Table 4 to Find Critical Values tc for a c Confidence Level 8.2 / 7 Using Table 4 to Find Critical Values of tc Find the column in the table with the given c heading Compute the number of degrees of freedom: d.f. = n 1 Read down the column under the appropriate c value until we reach the row headed by the appropriate d.f. If the required d.f. are not in the table: Use the closest d.f. that is smaller than the needed d.f. This results in a larger critical value tc. The resulting confidence interval will be longer and have a probability slightly higher than c. 8.2 / 8 Example: Find the critical value tc for a 95% confidence level for a t distribution with sample size n = 8. • Find the column in the table with c heading 0.950 8.2 / 9 Example cont. • Compute the number of degrees of freedom: d.f. = n 1 = 8 1 = 7 Read down the column under the appropriate c value until we reach the row headed by d.f. = 7 10 Example cont. tc = 2.365 11 PROCEDURE How to find Confidence Intervals for When is Unknown x E x E E tc s n c = confidence level (0 < c < 1) tc = critical value for confidence level c, and degrees of freedom d.f. = n 1 8.2 / 12 Example Confidence Intervals for , is Unknown The mean weight of eight fish caught in a local lake is 15.7 ounces with a standard deviation of 2.3 ounces. Construct a 90% confidence interval for the mean weight of the population of fish in the lake. Solution Mean = 15.7 ounces Standard deviation = 2.3 ounces. n = 8, so d.f. = n – 1 = 7 For c = 0.90, Appendix Table 4 gives t0.90 = 1.895. s 2.3 E tc 1.895 1.54 n 8 8.2 / 13 Example cont. E = 1.54 The 90% confidence interval is: 15.7 - 1.54 < < 15.7 + 1.54 14.16 < < 17.24 We are 90% sure that the true mean weight of the fish in the lake is between 14.16 and 17.24 ounces. 8.2 / 14 Summary: Confidence intervals for the mean Assume that you have a random sample of size n > 1 from an x distribution and that you have computed x and c. A confidence interval for μ is x E x E Where E is the margin of error. How do you find E? It depends of how of how much you know about the x distribution. 8.2 / 15 Summary cont. Situation 1 (most common) You don’t know the population standard deviation σ. In this situation, you use the t distribution with margin of error E zc s and degree of freedom d.f. = n – 1 n Although a t distribution can be used in many situations, you need to observe some guidelines. If n is less than 30, x should have a distribution that is mound-shaped and approximately symmetric. Its even better if the x distribution is normal. If n is 30 or more, the central limit theorem implies that these restrictions can be relaxed 8.2 / 16 Summary cont. Situation 2 (almost never happens!) You actually know the population value σ. In addition, you know that x has a normal distribution. If you don’t know that the x distribution is normal, then your sample size n must be 30 or larger. In this situation, you use the standard normal z distribution with margin of error E zc n 8.2 / 17 Summary cont. Which distribution should you use for Examine problem statement x ? (a) If σ is known use normal distribution with margin of error E zc n (b) If σ is not known use Student’s t distribution with margin of error E tc Assignment 20 s n d.f. = n - 1 8.2 / 18