Spring 2012 Chap 7

Spring 2012 Chap 7 - Chapter 7 In Chapter 6 we knew the...

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Chapter 7 In Chapter 6 , we knew the population standard deviation . 1 • Confidence interval for the population mean : 2 • Hypothesis test statistic for the population mean : 3 • Used the distribution In Chapter 7 , we don’t know the population standard deviation 1 2 • Use the sample standard deviation ( s ) 1 • Confidence interval for the population mean : 2 • Hypothesis test statistic for the population mean : 1.1.1.1 i.e. the test statistic, t is from the t-distribution with n-1 degrees of freedom. 1.1.1.2 3 • Sometimes you’ll see the symbol for standard error : When the standard deviation of a statistic is estimated from the data, the result is called the standard error of the statistic. Using the t-distribution : 1 • Suppose that an SRS of size n is drawn from a N ( , ) population. 1 • There is a different t-distribution for each sample size, so t(n-1) stands for the t distribution with n-1 degrees of freedom. 1 • Degrees of freedom = n – 1 = sample size – 1 1 • As the degrees of freedom increase, the t-distribution looks more like the normal distribution (because as n increases, s → σ ). t(n-1) distributions are symmetric about 0 and are bell shaped, they are just a bit wider than the normal distribution. STAT 301 Spring 2012 Chapter 7 Page 1
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2 1 • Table shows upper tails only, so if t * is negative, P(t < t * ) = P(t > |t * |) . if you have a 2-sided test, multiply the P(t > |t * |) by 2 to get the area in both tails. The normal table showed lower tails only, so the t -table is the reverse. 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 that appears on the table to be conservative. STAT 301 Spring 2012 Chapter 7 Page 2
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Example 1 : How accurate are radon detectors of a type sold to homeowners? To answer this question, university researchers placed 12 detectors in a chamber that exposed them to 105 picocuries per liter of radon. The detector readings were as follows: 91.9 97.8 111.4 122.3 105.4 95.0 103. 8 99.6 119.3 104.8 101.7 96.6 The sample mean = 104.13 and sample standard deviation, s = 9.40 of this data was calculated using SPSS. 1 a) Find a 90% confidence interval for the population mean STAT 301 Spring 2012 Chapter 7 Page 3
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Now re-do the above example using SPSS completely. Descriptives Statistic Std. Error Radon Mean 104.133 2.7128 90% Confidence Interval for Mean Lower Bound 99.261 Upper Bound 109.005 5% Trimmed Mean 103.804 Median 102.750 Variance 88.312 Std. Deviation 9.3974 Minimum 91.9 Maximum 122.3 Range 30.4 Interquartile Range 13.0 Skewness .853 .637 Kurtosis -.007 1.232 Steps for the One-Sample t test: 1. State the Null and Alternative hypothesis. 2. Find the test statistic: Suppose that an SRS of size n is drawn from a population having unknown mean µ. To test the hypothesis based on a SRS of size n, compute the one- sample t statistic 3. Calculate the p -value.
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