Transcript t statistic
PRED 354 TEACH. PROBILITY & STATIS. FOR PRIMARY MATH Lesson 8 Introduction to the t statistics t statistic Use sample mean to test hypotheses about a population. X X z 1. It is expected more or less to approximate its population mean (µ). 2. The standard error provides a measure of how well a sample mean approximates the population mean. 3. Compare sample mean with the hypothesized population mean by computing a z-score test statistic. t statistic: a substitute for z When the variability for the population is not known, sample variability is used in its place. SS SS s n 1 df s sX n 2 Estimated standard error It is used as an estimate of X when the value of σ is unknown. t statistic: a substitute for z X t sX Both z and t formulas are used for hypothesis testing. When you know the value σ2, use zscore. If σ2 is unknown, use t statistic. The larger the value of df is , the more closely the t distribution approximates a normal distribution. Determining probabilities for t distributions t distribution table is used to find probabilities for t statistics. TABLE…. Hypothesis tests with the t statistic Step 1. The hypotheses are stated, and the alpha level is set. Step 2. Critical region is located. Step 3. The sample data are collected, and the test statistic is computed. Step 4. The null hypothesis is evaluated. Example A professor of math hypothesizes that an introductory course in logic will help college student with their other studies. To test the hypothesis, a random sample of n = 25 freshman is selected. These students are required to complete a logic course during their freshman year. At the time of graduation, the final grade point average is computed for each of these students. The mean GPA for this sample X 2.83 with SS=6. Can the professor conclude that the grades for the sample were significantly different from the rest of the graduating class, which had an average GPA of µ=2.58? Test at α=.05. a. State the hypotheses. b. Determine the value of df, and locate the critical region. c. Compute the test statistic. d. Make a decision regarding H0. Example Fifteen years ago, the average weight of Turkish women the ages of 30 and 40 was µ=67 kg. A researcher would like to determine whether there has been any change in this figure during the past 15 years. A sample of n=100 women is obtained. The average weight for this sample is 61 with standard deviation of s=12. On the basis of these data, can the researcher conclude that there has been a significant change in weight? Use a two-tailed test with α=.01. Hypothesis tests with two independent samples: Assumptions 1. 2. The values in the sample must consist independent observations. The population sampled must be normal. of PRED 354 TEACH. PROBILITY & STATIS. FOR PRIMARY MATH Lesson 9 Hypothesis tests with two independent samples Hypothesis tests with two independent samples Population A Population B Sample B Sample A Hypothesis tests with two independent samples Sample statistic – population parameter t ( X 1 X 2 ) ( 1 2 ) t sX X obtained difference difference exp ected by chance 1 2 Estimated error sX X 1 2 s 2 p s 2 p n1 n2 Weighted variance (Pooled variance) SS1 SS2 s df1 df 2 2 p Degrees of freedom for the t statistic is df=df1+df2 Hypothesis tests with two independent samples Step 1. The hypotheses are stated, and the alpha level is set. Step 2. Critical region is located. Step 3. The sample data are collected, and the test statistic is computed. Step 4. The null hypothesis is evaluated. Examples In recent years, psychologist have demonstrated repeatedly that using mental images can greatly improve memory. A hypothetical experiment, designed to examine this phenomenon, is presented here. The psychologist first prepares a list of 40 pairs of nouns (for example, dog/bicycle, grass/door, lamp/piano). Next, two groups of subjects are obtained (two separate samples). Subjects in the first group are given the list for 5 minutes and instructed memorize the 40 noun pairs. Subjects in the second group receive the same list of words, but in addition to the regular instructions, these people are told to form a mental image for each pair of nouns (imagine a dog riding a bicycle, for example). Examples Later each group given a memory test, and the psychologist records the number of words correctly recalled for each individual. The data from this experiment are as follows. On the basis of these data, can the psychologist conclude that mental images affected memory? Number of words recalled Group 1 (No images) Group 2 (Images) 24, 23, 16, 17, 19, 13, 17, 20, 15, 26 18, 19, 23, 29, 30, 31, 29, 26, 21, 24 Hypothesis tests with two independent samples: Assumptions 1. The observations within each sample must be independent. 2. The two population from which the samples are selected must be normal. 3. The two populations from which the samples are selected must have equal variances.