# How is the Student’s t-distribution related the bell curve

## What are the similarities between the T and normal distributions?

Like the

**normal distribution**, the**t**–**distribution**is symmetric. If you think about folding it**in**half at the mean, each side will be the same. Like a standard**normal distribution**(or z-**distribution**), the**t**–**distribution**has a mean**of**zero. The**normal distribution**assumes that the population standard deviation is known.## What is the relationship between the standard normal distribution and the t distribution?

The mean of the

**distribution**is equal to 0. The variance is equal to ν/(ν − 2 ), if ν > 2. The variance is always greater than 1, although it is close to 1 when there are many degrees of freedom. With infinite degrees of freedom, the**t**–**distribution**is the same as the**standard normal distribution**.## What are the uses of Student t distribution?

**Student’s t**–

**distribution**or

**t**–

**distribution**is a probability

**distribution**that is used to calculate population parameters when the sample size is small and when the population variance is unknown.

## What is the basic shape of the Student t distribution?

The

**t**–**distribution**is symmetric and bell-shaped, like the normal**distribution**. However, the**t**–**distribution**has heavier tails, meaning that it is more prone to producing values that fall far from its mean.## Why does T distribution have fatter tails?

**T distributions have**a greater chance for extreme values than normal

**distributions**, hence the

**fatter tails**.

## What does S stand for in t distribution?

where x

**is the**sample**mean**, μ**is the**population**mean**,**s is the**standard deviation of the sample, and n**is the**sample size. The**distribution**of the**t**statistic is called the**t distribution**or the Student**t distribution**.## What happens to the T distribution as the sample size decreases?

The shape of the

**t distribution**changes with**sample size**. As the**sample size increases**the**t distribution**becomes more and more like a standard**normal distribution**. In fact, when the**sample size**is infinite, the two**distributions**(**t**and z) are identical.## What does the 95% represent in a 95% confidence interval?

Strictly speaking a

**95**%**confidence interval**means that if we were to take 100 different samples and compute a**95**%**confidence interval**for each sample, then approximately**95**of the 100**confidence intervals**will contain the true**mean**value (μ). Consequently, the**95**%**CI**is the likely range of the true, unknown parameter.## Why do we use t distribution?

The

**t**–**distribution**is most useful for small sample sizes, when the population standard deviation is not known, or both. As the sample size increases, the**t**–**distribution**becomes more similar to a normal**distribution**.## What does the T distribution tell us?

The

**t**–**distribution is**a way of describing a set of observations where most observations fall close to the mean, and the rest of the observations make up the tails on either side. It**is**a type of normal**distribution**used for smaller sample sizes, where the variance in the data**is**unknown.## What are the 3 characteristics of t distribution?

There are

**3 characteristics**used that completely describe a**distribution**: shape, central tendency, and variability.## What is the T critical value at a .05 level of significance?

**05**,) the

**t**crit

**value**is 1.895.

## What is the T critical value?

The

**t**–**critical value**is the cutoff between retaining or rejecting the null hypothesis. If the**t**-statistic**value**is greater than the**t**–**critical**, meaning that it is beyond it on the x-axis (a blue x), then the null hypothesis is rejected and the alternate hypothesis is accepted.## What is the z score for a 95% confidence interval?

The

**Z value**for**95**%**confidence**is**Z**=1.96.## What is the critical value for a 96 confidence interval?

Confidence Level | z |
---|---|

0.90 | 1.645 |

0.92 | 1.75 |

0.95 | 1.96 |

0.96 | 2.05 |

## What is the critical value of 99%?

Thus

**Z**_{α}_{/}_{2}= 1.645 for 90% confidence. 2) Use the t-Distribution table (Table A-3, p. 726). Example: Find**Z**_{α}_{/}_{2}for 98% confidence.Confidence (1–α) g 100% | Significance α | Critical Value Z_{α}_{/}_{2} |
---|---|---|

90% | 0.10 | 1.645 |

95% | 0.05 | 1.960 |

98% | 0.02 | 2.326 |

99% | 0.01 | 2.576 |

## What is the critical value of 95%?

The

**critical value**for a**95**% confidence interval is 1.96, where (1-0.95)/2 = 0.025.## What is the critical value of 86%?

**What is the critical**z-

**value**that corresponds to a confidence

**level of 86**%? approximately 1.48, 1.55 or 1.75.

## What is the critical value of 88%?

If we seek an

**88**% confidence interval, that means we only want a 12% chance that our interval does not contain the true**value**. Assuming a two-sided test, that means we want a 6% chance attributed to each tail of the**Z**-distribution. Thus, we seek the**z**α/2**value**of**z**0.06 .## What is the critical value of Z?

In this example, we observed

**Z**=2.38 and for α=0.05, the**critical value**was 1.645.Lower-Tailed Test | |
---|---|

a | Z |

0.10 | -1.282 |

0.05 | -1.645 |

0.025 | -1.960 |

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6 nov. 2017

## How do you find t critical value?

To

**find**a**critical value**, look up your confidence level in the bottom row of the table; this tells you which column of the**t**-table you need. Intersect this column with the row for your df (degrees of freedom). The number you**see**is the**critical value**(or the**t***-**value**) for your confidence interval.