Study l Guide x C Aplia: Student Quest x D cengageBrain My H x Aplia: Student Qu

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Study l Guide x C Aplia: Student Quest x D cengageBrain My H x Aplia: Student Quest x Y G critical value table servlet/quiz? quiz action takeQuiz&quiz; probGuid ONAPCOA801010000003700b830060000&ctx; frotella-0011&ck; 9 1493092593 3. The t statistic, the t distribution, and sample size Aa Aa The average age at which adolescent girls reach their adult height is 16 years. Suppose you have a sample of 27 adolescent girls who are developmentally delayed, and who have an average age at which they reached their adult height of 17.1 years and a sample variance of 36.0 years. You want to test the hypothesis that adolescent girls who are developmentally delayed have a different age at which they reached their adult height than all adolescent girls. Calculate the t statistic. To do this, you first need to calculate the estimated standard error. The estimated standard error is SM The t statistic is Now suppose you have a larger sample size n 81. Calculate the estimated standard error and the t statistic for this sample with the same sample average and the same standard deviation as above, but with the larger sample size. The new estimated standard error is The new t statistic is Note that the t statistic becomes as n becomes larger. Use the Distributions tool to look at the t distributions for different sample sizes. To do this, choose the Degrees of Freedom for the first sample size on the slider, and click the radio button with the single orange line. Move the orange vertical line to the right until the number below the orange line is located on the t statistic. The probability of getting that t statistic or one more extreme will appear in the bubble with the orange type. Now repeat the process for the other sample. t Distribution Degrees of freedom 52

Explanation / Answer

The estimated standard error sM=sqrt(variance/n)=sqrt(36/27)= 1.1547

Test statistic t=(Mean-16)/sM=(17.1-16)/1.1547=0.9526

For sample size n=81

The estimated standard error sM=sqrt(variance/n)=sqrt(36/81)= 0.6667

Test statistic t=(Mean-16)/sM=(17.1-16)/0.6667=1.6499

The t-statistic becomes larger as n become larger.

First case df=n-1=27-1=26

Second case df=81-1=80

For n=27, p-value=0.3496

For n=81, p-value=0.1029

The t distribution is fatter tailed with smaller n

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