You are watching: Why do t distributions tend to be flatter
UNKNOWNZ-scores are offered when the population standard deviation (sigma) is known. The t statistic uses the sample variance or standard deviation in place of the unknown populace values.
The sample variance (s^2) in the t formula changes from one sample come another and contributes to the variability of the t statistic.A z-score uses the populace variance which is continuous from one sample come another.Normal distributions space based on big sample sizes while t-distributions are based upon smaller sample sizes.
If other factors are organized constant, describe howeach the the following influences the value of theindependent-measures t statistic and also the likelihoodof rejecting the null hypothesis:a. Boosting the number of scores in every sample.b. Boosting the variance for each sample.
A) The dimension of the two samples influences the size of the estimated standard error in the denominator of the t-statistic. As sample size increases, the value of t additionally increases (moves farther away from zero), and also the likelihood the rejecting H0 also increases.B) The variability the the score impacts the approximated standard error in the denominator that the t-statistic. As the variability of a scores rises the worth of t to reduce (becomes closer to zero) and also the likelihood the rejecting the H0 decreases.
Describe the homogeneity of variance assumptionand describe why the is important for the independent steps t test.
The homogeneity of variance assumption specifies that the variances room equal for the two populations from which the samples are obtained.If this presumption is violated, the t-statistic can cause misleading conclusions for a theory test.
The shortcoming of using a z-score because that hypothesis testing is that the z-score formularequires an ext information 보다 is commonly available.Specifically, a z-score calls for thatwe know the value of the population standard deviation (or variance), which is neededto compute the conventional error.In most situations, however, the traditional deviation for the population is no known.
The z-score provides the actual populace variance, σ^2 (or the conventional deviation)The t formula uses the matching sample variance (or conventional deviation) as soon as the population value is no known.
Describe the number of scores in a sample that are independent and free to vary.Because the sample mean locations a limit on the value of one score in the sample, there are n - 1 degrees of flexibility for a sample through n score.The higher the value of df because that a sample, the much better the sample variance, s^2, representsthe population variance, σ^2, and the much better the t statistic almost right the z-score.How fine a t distribution approximates a regular distributor is identified by levels of freedom.In general, the greater the sample dimension (n) is, the larger the levels of flexibility (n - 1) are,and the better the t circulation approximates the normal distribution.
For both formulas, z and also t, the optimal of the formula, M - μ, have the right to take on various valuesbecause the sample mean (M) varies from one sample to another. For z-scores, however,the bottom the the formula does not vary, listed that every one of the samples space thesame size and are selected indigenous the same population.Specifically, every one of the z-scores have actually the very same standard error in the denominator, M 2/n, since the population variance and the sample dimension are the same for every sample.For t statistics the bottom that the formula different from one sample come another
Increasing sample size boosts the likelihood the rejecting the null hypothesis but has littleor no result on steps of impact size.
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How go the standard deviation affect the outcome of a hypothesis test and measures of result size?
Increasing the sample variance reduce the likelihood that rejecting the null hypothesis and also reduces procedures of result size.
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