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Lesson 10.2.1

The T-Distribution and T-Statistics

STATWAY™STUDENT HANDOUT

Lesson 10.2.1

The T-Distribution and T-Statistics

STUDENT NAME / DATE

Introduction

Previously, we have seen that the sampling distribution of sample means is normally distributed. As with any normal distribution, we are able to standardize a distribution of sample means by taking Z-scores. We are able, that is, if we know the population standard deviation. But what happens when we don’t know it? In this lesson, we will confront the reality that population standard deviations are rarely known, and this influences how we do statistical inference regarding a population mean.

When σ is Known

Suppose we have a population whose standard deviation,σ, is known. With a sufficiently large sample size, a sampling distribution of sample means is approximately normal. The standard error of the sampling distribution is given by:

Given a particular sample mean from a sample of size , the standardized value (Z-score) of the sample mean is called the Z-test statistic. This test statistic is computed with the formula

When the criterion for the normality of the distribution ofsample means is met (, or the population from which we are sampling is normal), the distribution of sample means may be considered normal, and the distribution ofZ statistics above is normal as well.

We have seen with sampling distributions of sample proportions, the sampling distribution and test statistic can be used to construct confidence intervals and conduct hypothesis tests about the population mean.

When σ is Unknown

In most situations when we seek to estimate the population mean, we do not know the population standard deviation. That is, σ is unknown. The only option available is to approximate σ with asample standard deviation. When we make this substitution, the standard error of the sampling distribution is estimated by

Since s is only an approximation of σ, the standard error estimate creates additional variability to the test statistic. This added variability means that the distribution of suchtest statistics,

isnot normal. They instead varyaccording to Student’s T-distribution. (The distribution is named after researcher William Gosset, whose pen name was Student.)

The T-Distribution

The T-distribution describes the variability of the T-test statistic,

The T-distribution exists because of the sample standard deviation that is used in the denominator of the test statistic. The sample standard deviation varies from sample to sample, and the smaller the sample size, the more it varies, and the wider the T-distribution gets.

The width of the T-distribution depends on how much a sample standard deviation can vary. The amount of variability depends on the number of deviations that can vary freely in the computation of the sample standard deviation. The fewer the random deviations are, the more the sample standard deviation varies.

It is not hard to show that the deviations from the mean, , which are averaged in a sample standard deviation always add to zero. Because of this, the last deviation summed in this average is not a free, random variable – it is always the value that makes the resulting sum zero. While there are deviations from the mean, only of these are free random variables.

Thus, the variability of standard deviations depends on the number of free random deviations in the sample standard deviation, , and so this quantity is known as the degrees of freedom. Technically speaking, each degree of freedom defines a uniquely associated T-distribution.

TheT-distribution familyis a collection of continuous probability distributions with the following characteristics:

  • T-distributions are bell-shaped and symmetric with a mean of 0.
  • Each distribution depends on the degrees of freedom, d.f.
  • T-distributions have heavier tails and narrower peaks than the standard normal distribution.
  • The area under each T-distribution curve is 1.
  • As the degrees of freedom increase, the tails are lighter, and T-distribution eventually approaches the standard normal distribution.
  • When making inferences about a population mean, the degrees of freedom are equal to the sample size minus 1 (d.f. = n – 1).

Similar to the Z-statistic, the T-statistic is anestimate of how many standard errors the sample mean is from the hypothesized population mean.

Try These

Due to rising sea temperatures resulting from global warming, polar bears are in jeopardy of becoming an endangered species. The leading cause of concern is the melting of sea ice that polar bears use for hunting and dens. The erosion of their habitat creates smaller seal populations, resulting in a lack of food for the bears. Biologists regularly visit Arctic regions to track the health and numbers of polar bears.

Biologists estimate that the average weight of an adult male polar bear is approximately 475 kg (1050 lbs.).Assume that polar bear weights are normally distributed.

Suppose that four random samples of 5 polar bears are drawn from four different areas in Alaska. The weights of the polar bears (in kilograms) are displayed in the table below.

Sample A / 466 / 520 / 512 / 513 / 498
Sample B / 471 / 476 / 461 / 453 / 423
Sample C / 493 / 482 / 431 / 450 / 452
Sample D / 481 / 492 / 475 / 498 / 512

1All of the samples have the same sample size. What is the appropriate number of degrees of freedom for this sample size?

2Calculate the means and standard deviations for each sample. Complete the table below:

Mean / Sample Standard Deviation
Sample A / 501.8 / 21.54
Sample B / 456.8 / 20.88
Sample C / 461.6 / 25.32
Sample D / 491.6 / 14.53

3Calculate the T-statistic for each sample, assuming that the population mean weight is µ = 475 kg. Complete the table below.

T-Statistic
Sample A / 2.78
Sample B / -1.94
Sample C / -1.18
Sample D / 2.55

4Which of the four T-statistics indicates a sample mean that is the farthest away (in estimated standard errors) from the population mean µ = 475?

5Which of the four T-statistics indicates that the sample mean(s) are below µ = 475?

6Which of the four T-statistics indicates a sample mean that is the closest (in estimated standard errors) to µ = 475?

7Based on the sample statistics, which sample(s) should be most concerning to biologists and conservationists?

take It Home

Are snakes left-handed or right-handed?

Sometimes snakes coil so that their left side is pointing in and sometimes they coil so that their right side is pointing in. While watching a snake for a period of time, you can count the number of times a snake coils left and the number of times it coils right. For the following experiment, several cottonmouth snakes were observed over a period of time[1]. And for each snake, the proportion of times it coils to the left was recorded. If a snake coils to the left half the time the proportion would be 0.5. If it coiled to the left more than half the time the proportion would be greater than 0.5, and if it coiled to the left less than half the time the proportion would be less than 0.5.

Four different categories of snakes were observed: adult males, adult females, juvenile males, and juvenile females. Several snakes in each category were observed, and the proportion of left coils was recorded for each. Assume that the measure “proportion of left coils” is normally distributed.

Part 2 – Adult Females

Below are the left-coil proportions for 15 adult females:

0.582 / 0.585 / 0.550 / 0.554 / 0.609
0.545 / 0.544 / 0.600 / 0.638 / 0.656
0.600 / 0.696 / 0.424 / 0.493 / 0.491

1How many degrees of freedom are associated with this sample size?

2Find the sample mean and sample standard deviation.

3Assuming that µ = 0.5, calculate the T-statistic for the given sample.

4What does this test statistic imply about this sample of 15 adult females?

5What does this test statistic suggest about the population of adult females’ coiling preference?

Part 2 – Adult Males, Juvenile Males, Juvenile Females

The coiling data for three other categories of snakes are shown below.

Adult males / 0.563 / 0.556 / 0.522 / 0.541 / 0.395
Juvenile males / 0.486 / 0.492 / 0.475 / 0.464 / 0.493
Juvenile females / 0.512 / 0.556 / 0.565 / 0.417 / 0.429

1Each sample has the same sample size. What are the appropriate degrees of freedom for each?

2Calculate the mean and standard deviations for each sample.

3Calculate the T-statistics for each sample, assuming a population mean of µ = 0.5. Complete the table below.

4Which of the three T-statistics indicates that the sample mean is farthest away from the population mean µ = 0.5?

5Which T-statistics indicate that the sample means are below µ = 0.5?

6Which of the three T-statistics indicates that the sample mean is closest to µ = 0.5?

7Based on the sample statistic, which group of snakes appears to most favor a left-coiling? Which group appears to most favor right-coiling?

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This lesson is part of STATWAY™, A Pathway Through College Statistics, which is a product of a Carnegie Networked Improvement Community that seeks to advance student success. Version 1.0, A Pathway Through Statistics, Statway™ was created by the Charles A. Dana Center at the University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching. This version 1.5 and all subsequent versions, result from the continuous improvement efforts of the Carnegie Networked Improvement Community. The network brings together community college faculty and staff, designers, researchers and developers. It is an open-resource research and development community that seeks
to harvest the wisdom of its diverse participants in systematic and disciplined inquiries to improve developmental mathematics instruction. For more information on the Statway Networked Improvement Community, please visit carnegiefoundation.org. For the most recent version of instructional materials, visit Statway.org/kernel.

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A Pathway through statistics, version 1.5, STATWAY™ - STUDENT HANDOUT

[1] [Ref: Roth, E. D. (2003). ‘Handedness’ in snakes? Lateralization of coiling behavior in a cottonmouth, Agkistrodonpiscivorusleucostoma, population. Animal Behaviour, 66, 337-341.]