Chapter 7

# Chapter 7 - 1 Chapter 7 Section 7.1 Inference for the Mean...

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Unformatted text preview: 1 Chapter 7 Section 7.1: Inference for the Mean of a Population Section 7.2: Comparing Two Means Suppose that we have a simple random sample (SRS) of size n drawn from a normal population: N ( μ , σ ). Suppose also that μ and σ are unknown parameters. If we are interested in using inference techniques with data sets of this type, we will need a different set of tools from those we used in chapter 6. In Chapter 6 , we knew the population standard deviation σ . • Confidence interval for the population mean μ : * X x z n σ ± • Hypothesis test statistic for the population mean μ : / x z n μ σ- = • Used the standard normal distribution. If we don’t know σ , the population standard deviation, then we will have to use our best estimate of σ , the sample standard deviation s. Using s instead of σ means that we are no longer able to use the standard normal distribution. Instead, we will have to use the student’s t distribution. The student’s t distribution is completely determined by the number of degrees of freedom. When looking at the distribution of X , we will use the t distribution with n-1 degrees of freedom, the t(n-1) distribution. 2 Using the t distribution • Suppose that an SRS of size n is drawn from a N ( μ , σ ) population. • There is a different t distribution for each sample size, so t(k) stands for the t distribution with k degrees of freedom. • Degrees of freedom = k = n – 1 = sample size – 1 • As k increases, the t distribution looks more like the normal distribution (because as n increases, s → σ ). t(k) distributions are symmetric about 0 and are bell shaped, they are just a bit wider than the normal distribution. • The t table shows upper tails only, so o if t * is negative, P(t < t * ) = P(t > |t * |) . o if you have a 2-sided test, multiply the P(t > |t * |) by 2 to get the area in both tails. o The normal table showed lower tails only, so the t-table is backwards. 3 4 The One-Sample t Confidence Interval : * s x t n ± where t* is the value for the t ( n-1) density curve with area C between – t* and t* . Finding t* on the table: Start at the bottom line to get the right column for your confidence level, and then work up to the correct row for your degrees of freedom. What happens if your degrees of freedom isn’t on the table, for example df = 79? Always round DOWN to the next lowest degrees of freedom to be conservative. Example 1 a) Find t* for an 80% confidence interval if the sample size is 20. b) Find t* for an 98% confidence interval if the sample size is 35. c) Find a 95% confidence interval for the population mean if the sample mean is 42 and the sample size is 50. 5 The One-Sample t test : • State the Null and Alternative hypothesis....
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Chapter 7 - 1 Chapter 7 Section 7.1 Inference for the Mean...

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