AP Statistics

 If our data comes from a simple random sample (SRS) and the sample size is sufficiently large, then we know that the sampling distribution of the sample means is approximately normal with mean μ and standard deviation.  The spread of the sampling distribution depends on n and σ. σ is generally unknown and must be estimated.  NOW…THEORY ASIDE AND ONTO PRACTICE ! AP Statistics, Section 11.12

 SRS – size n  Normal distribution of a population  μ and σ are unknown  To estimate σ – use “S” in its place Then the standard error of the sample mean is AP Statistics, Section 11.13

 The z statistic has N (0,1)  When s is substituted the distribution is no longer normal AP Statistics, Section 11.14

 The t statistic is used when we don’t know the standard deviation of the population, and instead we use the standard deviation of the sample distribution as an estimation.  The t statistic has n-1 degrees of freedom (df). AP Statistics, Section 11.15

 Interpret the t statistic in the same way as the z statistic  There is a different distribution for every sample size.  The t statistic has n-1 degrees of freedom.  Write t (k) to represent the t distribution with k degrees of freedom. AP Statistics, Section  Density curves for the t distribution are similar to the normal curve (symmetrical and bell shaped)  The spread is greater and there is more probability in the tails and less in the center.  Using s introduces more variability than sigma.  As d.f. increase, t(k) gets more normal

 In statistical tests of significance, we still have H 0 and H a.  We need to provide the mu in the calculation of the t statistic.  Looking at the t table is fundamentally different than the z table. AP Statistics, Section 11.17

 Assume SRS size n with population mean μ  Confidence interval will be correct for normal populations and approx. correct for large n. AP Statistics, Section 11.18

 Let’s suppose that Mr. Young has been told that he should mop the floor by 1:25 p.m. each day.  We collect 12 sample times with an average of minutes after 1 p.m. and with a standard deviation of minutes.  Find a 95% confidence interval for Mr. Young’s mopping times. AP Statistics, Section 11.19

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Step 1:  Population of interest: ◦ Mr. Young’s mopping time  Parameter of interest: ◦ average time of arrival to mop  Hypotheses ◦ H 0 : µ=25 min past 1:00 ◦ H a : µ>25 min past 1:00 AP Statistics, Section

 We are using 1 sample t-test?  Bias? ◦ SRS not stated. Proceed with caution.  Independence? ◦ Population size is at least 10 times the sample size? ◦ We assume that Mr. Young has mopped on a lot of days  Normality? ◦ Big sample size (> 30). No ◦ Sample is somewhat normal because the sample distribution is single peaked, no obvious outliers. AP Statistics, Section

 Calculate the test statistic, and calculate the p-value from Table C AP Statistics, Section

 Is the t-value of statistically significant at the 5% level? At the 1% level?  Does this test provide strong evidence that Mr. Young arrives on time to complete his mopping? AP Statistics, Section Try this exercise on your calculator using: STAT TESTS Tinterval STAT TESTS T-Test

 Wednesday: 11.6 –  Thursday:11.13 –  Friday:T-Test Worksheet AP Statistics, Section