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Lecture

Ch 9- The t Distributions.docx

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Department
Psychology
Course
PSYC 1010
Professor
Hank Davis
Semester
Summer

Description
The t Distributions  The t distributions are used when we don’t have enough information to use the z distribution.  We have to use the t distribution when we don’t know the population standard deviation or when we compare 2 samples to one another  As the sample size gets bigger, the t distributions begin to merge with the z distribution because we gain confidence as more and more participants are added to our study Estimating Population Standard Deviation from the Sample Using sample ID to Estimate population SD  We don’t know the true value of the population standard deviation (σ).  Our best estimate of this population parameter comes from our sample statistic (i.e., estimate population SD using sample SD)  Recall that any sample will be less variable than its parent population  Without correction for this difference, sample variance and SD would give us an underestimate of population variability.  When calculating sample variance and standard deviation, use n-1 instead of n to adjust for the bias in sample variability 2 å (X - m) s = N  Population SD:  Sample SD: 2 å (X - M) s = n-1  Step 1: calculate the sample mean  Step 2: Use this sample mean in the corrected formula for the standard deviation Calculating Standard Error for the t statistic  In a z test, you compare your sample to a population with a known mean and standard deviation.  In actual research practice, you often compare two or more groups of scores to each other, without any direct information about the populations.  Often, nothing is known about the populations that the samples are supposed to come from.  The t statistic: o When the variability of the population is unknown (s), we use the t- statistic. o The t-statistic will let us substitute sample variance (or standard dev.) for population variance. o This means we use ‘s’ instead of ‘σ’ o Using the sample variance we can estimate the standard error of the mean. o Instead of: use or o The estimated standard error (sM) is used as an estimate of the real standard error (M ) when the unknown. o It is computed from the sample variance or sample standard deviation and provides an estimate of the standard distance between a sample mean(m) and the Population mean (µ). Using Standard error to calculate the t statistic  The t-statistic is used to test hypotheses about an unknown population mean (μ) when the value of σ is unknown.  T statistic: indicates the distance of a sample mean from the population mean in terms of the standard error  The formula for the t-statistic has the same structure as the z-score formula, expect the t-statistic uses estimated standard erMor (s ) in the denominator. M-µ t = sM  Formula:  Using the z test, you learned that when the population distribution follows a normal curve, the shape of the distribution of means will also be a normal curve. However, this changes when you do hypothesis testing with an estimated population variance.  Because our estimate of σ is based on our sample (s ), and because from 2 sample to sample our estimate of σ will change…there is more variation in our estimate of σ , and more variation in the t distribution than in a z distribution.  Like the normal distribution, the t-distributions are bell shaped, symmetrica land have a mean of 0.  However, t-distributions generally have more variability than the normal distribution (flatter and more spread out).  The t-distribution is really a “family” of distributions, as the exact shape changes with the degrees of freedom. The single-sample t Test  single-sample test: a hypothesis test in which we compare data from one sample to a population for which we know the mean but not the standard deviation The t Table and Degrees of Freedom  Degrees of freedom: the number of scores that are free to vary when estimating a population parameter from a sample.  Just how much the t distribution differs from the normal curve depends on the degree s of freedom.  The t distribution differs most from the normal curve when the degrees of freedom are low (because the estimate of the population variance is based on a very small sample).  In general, the greater the sample size (n), the larger the degrees of freedom (n-1), and the better the t-distribution will approximate the normal distribution.  Degrees of freedom describe the number of scores in a sample that are free to vary.  For each parameter you estimate, you lose one degree of freedom.  When we use the t statistic, because the known sample mean is used to replace the unknown population mean in calculating the estimated standard deviation, one degree of freedom is lost.  Thus, df= n-1.  This difference in shape across t-distributions (depending on degrees of freedom) affects how extreme a score you need to reject the null hypothesis.  As in the z-test, to reject the null hypothesis using a single sample t-test, your sample mean has to be in an extreme section of the distribution of means.  If the distribution has more of its means in the tails than a normal curve would have, then the point where the rejection region begins has to be further out on the comparison distribution.  Thus, it takes a slightly more extreme sample mean to get a significant result when using a t distribution than when using a standard normal curve.  As your sample size approaches infinity, the t distribution approaches the norma
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