$t(x)=\frac{\overline{X}-\mu}{\sqrt{\frac{1}{n}\sum {({X}_{i}-\overline{X})}^{2}}}\sqrt{n-1}$

$t(x)=\frac{\overline{X}-\mu}{\sqrt{\frac{1}{n-1}\sum {({X}_{i}-\overline{X})}^{2}}}\sqrt{n}$

Jayvion Caldwell
2022-07-22
Answered

Question about t statistic. Of these two, which formula is correct?

$t(x)=\frac{\overline{X}-\mu}{\sqrt{\frac{1}{n}\sum {({X}_{i}-\overline{X})}^{2}}}\sqrt{n-1}$

$t(x)=\frac{\overline{X}-\mu}{\sqrt{\frac{1}{n-1}\sum {({X}_{i}-\overline{X})}^{2}}}\sqrt{n}$

$t(x)=\frac{\overline{X}-\mu}{\sqrt{\frac{1}{n}\sum {({X}_{i}-\overline{X})}^{2}}}\sqrt{n-1}$

$t(x)=\frac{\overline{X}-\mu}{\sqrt{\frac{1}{n-1}\sum {({X}_{i}-\overline{X})}^{2}}}\sqrt{n}$

You can still ask an expert for help

Franklin Frey

Answered 2022-07-23
Author has **15** answers

They are mathematically equivalent, because

$\frac{\sqrt{n-1}}{\sqrt{1/n}}=\sqrt{n}\sqrt{n-1}=\frac{\sqrt{n}}{\sqrt{1/(n-1)}}.$

That said, the second formula is a better reflection of how the t statistic is calculated, because it is obtained by estimating the standard deviation of the population from the sample when it is unknown. Thus,

${s}^{2}=\frac{1}{n-1}\sum _{i=1}^{n}({X}_{i}-\overline{X}{)}^{2}$

is the unbiased estimator for the variance ${\sigma}^{2}$ when ${X}_{i}$ are iid normal random variables with unknown mean μ and unknown variance ${\sigma}^{2}$. Then the statistic

$\frac{X-\mu}{s/\sqrt{n}}$

is Student $t$ distributed where $s/\sqrt{n}$ is the standard error of the mean.

$\frac{\sqrt{n-1}}{\sqrt{1/n}}=\sqrt{n}\sqrt{n-1}=\frac{\sqrt{n}}{\sqrt{1/(n-1)}}.$

That said, the second formula is a better reflection of how the t statistic is calculated, because it is obtained by estimating the standard deviation of the population from the sample when it is unknown. Thus,

${s}^{2}=\frac{1}{n-1}\sum _{i=1}^{n}({X}_{i}-\overline{X}{)}^{2}$

is the unbiased estimator for the variance ${\sigma}^{2}$ when ${X}_{i}$ are iid normal random variables with unknown mean μ and unknown variance ${\sigma}^{2}$. Then the statistic

$\frac{X-\mu}{s/\sqrt{n}}$

is Student $t$ distributed where $s/\sqrt{n}$ is the standard error of the mean.

asked 2022-05-09

Let $\mathcal{X}$ be a sample space, $T$ a test statistic and $G$ be a finite group of transformations (with M elements) from $\mathcal{X}$ onto itself. Under the null-hypothesis the distribution of the random variable $X$ is invariant under the transformations in $G$. Let

$\hat{p}=\frac{1}{M}\sum _{g\in G}{I}_{\{T(gX)\ge T(X)\}}.$

Show that $P(\hat{p}\le u)\le u$ for $0\le u\le 1$ under the null hypothesis

$\hat{p}=\frac{1}{M}\sum _{g\in G}{I}_{\{T(gX)\ge T(X)\}}.$

Show that $P(\hat{p}\le u)\le u$ for $0\le u\le 1$ under the null hypothesis

asked 2020-10-25

If

asked 2021-02-11

Birth Weights Are the birth weights described quantitative data or categorical data?

asked 2021-05-21

A proposed study design is to leave 100 questionnaires by the checkout line in a student cafeteria. The questionnaire can be picked up by any student and returned to the cashier. Explain why this volunteer sample is a poor study design.

asked 2021-05-13

A boy is to sell lemonade to make some money to afford some holiday shopping. The capacity of the lemonade bucket is 1. At the start of each day, the amount of lemonade in the bucket is a random variable X, from which a random variable Y is sold during the day. The two random variables X and Y are jointly uniform.

Find and sketch the CDF and the pdf of Z which is the amount of lemonade remaining at the end of the day. Clearly indicate the range of Z

Find and sketch the CDF and the pdf of Z which is the amount of lemonade remaining at the end of the day. Clearly indicate the range of Z

asked 2022-01-16

What are the mean and standard deviation of a binomial probability distribution with n=169 and $p=\frac{1}{13}$ ?

asked 2022-03-01

Suppose that the height, in inches, of a 20-year-old man is a normal random variable

with parameters μ = 175 and **σ **= 10.

There are ten 20-year-old men in the class. What's the probability that more than 8

of them are over 185cm?