Problem of the Day. 7.3 Hypothesis Testing for Mean (Small Samples n<30) Objective(s): Find critical values in a t-distribution

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1 Problem of the Day Find the standardized test statistic (z) if the sample mean is 14, the standard deviation is 2, the sample size is 36, and the population mean is Hypothesis Testing for Mean (Small Samples n<30) Objective(s): Find critical values in a t-distribution Use the t-test to test a mean,μ, when n < 30 1

2 When we do not have at least 30 samples for our statistical test, we cannot use the standard normal distribution (z distribution) at the front of our books. The curve for these distributions with smaller sample sizes will still be symmetrical about the mean and bellshaped, but they will have thicker tails than the standard normal distribution curve. These are called t distributions. t distribution: a sampling distribution from less than 30 samples where a variable, x, is approximately normal. * There is not one curve for t distributions, but rather there is a family of curves. The curve shape will depend on the number of samples. * As the number of samples increase, the closer the curve will get to the standard normal curve. At 30 samples, the curve becomes the standard normal curve. 5 samples 10 samples 20 samples 30 samples standard normal (z) curve standardized test statistic for t distribution: t = x μ s/ n x = sample mean μ = population mean s = sample standard deviation n = number of samples degrees of freedom: the number of values that are free to vary after x is calculated. d.f. = n 1 "Now, imagine a set of three numbers, whose mean is 3. There are lots of sets of three numbers with a mean of 3, but for any set the bottom line is this: you can freely pick the first two numbers, any number at all, but the third (last) number is out of your hands as soon as you picked the first two. Say our first two numbers are... 1 and 6, giving us a set of two freely picked numbers, and one number that we still need to choose, x: [1, 6, x]. For this set to have a mean of 3, we don t have anything to choose about x. X has to be 2, because ( ) / 3 is the only way to get to 3. So, the first two values were free for you to choose, the last value is set accordingly to get to a given mean. This set is said to have two degrees of freedom, corresponding with the number of values that you were free to choose (that is, that were allowed to vary freely). This generalizes to a set of any given length. If I ask you to generate a set of 4, 10, or 1,000 numbers that average to 3, you can freely choose all numbers but the last one. In those sets the degrees of freedom are respectively, 3, 9, and 999. The general rule then for any set is that if n equals the number of values in the set, the degrees of freedom equals n 1." Ron Dotsch: Social Psychologist, Degrees of Freedom Tutorial of freedom/ accessed 11/3/2014 2

3 t distribution chart found in the very back of your book. α, level of significance makes up columns according to whether the test is one tailed (Right or left) or two tailed. degrees of freedom make up the rows * If test is left tailed (H a contains <), add a negative sign to the number found in the table. " " * If the test is right tailed (H a contains >), leave the number found in the table positive. "+" * If the test is 2 tailed (H a contains ) add a plus and minus sign to the number found in the table. "±" Finding Critical Values (t 0 ) in a t distribution (sample is less than 30) 1. Identify α 2. Identify the degrees of freedom d.f.=n 1 3. Find the critical values, t 0, using the t distribution table * use the Row with n 1 degrees of freedom * use α for the appropriate Column: if Left Tailed use the column titled "One Tail,α" & add a negative sign to number if Right Tailed use the column titled "One Tail,α" & keep number positive if Two Tailed use the column titled "Two Tails,α" & add a negative and a positive sign to number (±) 3

4 Find the critical value t 0 of a left tailed test given n= 21 and α=0.05. Find the critical value t 0 of a right tailed test given n= 17 and α=0.01. Find the critical values t 0 and t 0 of a two tailed test given n= 26 and α=0.05. Using t test for mean μ 1. State claim, H 0, H a 2. Identify α 3. Identify d.f.=n 1 4. Determine critical values, t 0, and rejection regions use d.f. for row use α for column with appropriate signs & Tail Test 5. Find t t= x μ s/ n 6. Make decision and interpret in context with the problem. 4

5 A used car dealer says that the mean price of a 1999 Ford F 150 Super Cab is at least $16,500. You suspect this claim is incorrect and find that a random sample of 14 similar vehicles has a mean price of $15,700 and a standard deviation of $1250. Is there enough evidence to reject the dealer's claim at α=0.05? Assume the population is normally distributed. There is sufficient evidence to reject the dealer's claim that the truck is at least $16,500. An industrial company claims that the mean ph level of the water in a nearby river is 6.8. You randomly select 19 water samples and measure the ph of each. The sample mean and standard deviation are 6.7 and 0.24, respectively. Is there enough evidence to reject the company's claim at α=0.05? Assume the population is normally distributed. 7.3: pg. 356: 3 13 odd, 17 20, There is insufficient evidence to reject the claim that the ph level is