Topic 2. Distributions and hypothesis testing [S&T Chap. 4-5]



Topic 2. Distributions, hypothesis testing, and sample size determination

2. 1. The Student - t distribution (ST&D pp56 and 77)

Consider a repeated drawing of samples of size n from a normal distribution. For each sample compute[pic], s, [pic], and another statistic, t, where

t(n-1) = ([pic] - ()/[pic]

The t statistic is the deviation of a normal variable [pic] from its hypothesized mean measured in standard error units. Phrased another way, is the number of standard errors that separate[pic]and µ. The distribution of this statistic is known as the Student's t distribution. There is a unique t distribution for each value of n. For example, for sample size 5, the corresponding t distribution is said to have (5 – 1) degrees of freedom. (df).

Now consider the shape of the frequency distribution of the sampled t values. Though symmetric and appearing quite similar to a normal distribution of sample means ([pic]), the t distribution will be more variable (i.e. larger dispersion, or broader peak) than Z because [pic] varies from sample to sample. The larger the sample size, the more precisely [pic] will estimate[pic], and the closer t approaches Z. t values derived from samples of size n ≥ 60 are approximately normally distributed. As n ( ∞, t ( Z.

[pic]

Fig. 1. Distribution of t (df=5-1=4) compared to Z. The t distribution is symmetric and somewhat flatter than Z, lying under it at the center and above it in the tails. The increase in the t value relative to Z is the price we pay for being uncertain about the population variance

2. 2. Confidence limits [S&T p. 77]

Suppose we have a sample {Y1, ..., Yn} with mean [pic] drawn from a population with unknown mean μ, and we want to estimate μ. If we manipulate the definition for the t statistic we get:

[pic] = ( ( t (n-1) [pic]

Note that μ is a fixed but unknown parameter while[pic] is a known but random statistic. The statistic [pic] is distributed about μ according to the t distribution; that is, it satisfies

Pr{[pic]- t (/2, n-1 [pic] ( ( ( [pic]+ t (/2, n-1 [pic]}= 1- (

Note that for a confidence interval of size α, you must use a t value corresponding to an upper percentile of α /2 since both the upper and lower percentiles must be included (see Fig. 1). Therefore the confidence interval is

[pic]- t (/2, n-1 [pic] ( ( ( [pic]+ t (/2, n-1 [pic]

The two terms on either side represent the lower and upper (1- () confidence limits of the mean. The interval between these terms is called the confidence interval (CI). For example, in the barley malt extract dataset (see SAS example below), [pic]= 75.94 and [pic] = 1.227 / √14 = 0.3279. Table A3 gives the t (0.025, df=13) value of 2.16, which we multiply by 0.3279 to conclude at the 5% level that μ = 75.94 ± 0.71.

μ = [pic] ± [pic] = 75.94 ± 2.16(0.3279) = 75.94 ± 0.71

It is incorrect to say that there is a probability of 95% that the true mean is within 75.94 ± 0.72. If we repeatedly obtained samples of size 14 from the population and constructed these limits for each, we could expect 95% of the intervals to contain the true mean.

True mean

Figure 2. The vertical lines represent 20 95% confidence intervals. One out of 20 intervals does not include the true mean (horizontal line). The confidence level represents the percentage of time the interval covers the true (unknown) parameter value [ST&D p.61]

2. 3. Hypothesis testing [ST&D p. 94]

Another use of the t distribution, more in the line of experimental design and ANOVA, is in hypothesis testing. Results of experiments are usually not clear-cut and therefore need statistical tests to support decisions between alternative hypotheses. Recall that in this case a null hypothesis Ho is established and an alternative H1 is tested against this. For example, we can use the barley malt data (ST&D p. 30) to test the Ho: μ = 78 against H1: μ ( 78.

DATA barley;

INPUT extract @@;

CARDS;

77.7 76.0 76.9 74.6 74.7 76.5 74.2 75.4 76.0 76.0 73.9 77.4 76.6 77.3

;

PROC TTEST h0=78 alpha=0.05;

var extract;

RUN; QUIT;

Statistics

Lower Upper

Variable N CI mean Mean CI mean Std Dev Std Err

extract 14 75.234 75.943 76.651 1.2271 0.3279

Variable DF t Value Pr > |t|

extract 13 -6.27 1.44. To converge better on the desired values, we need to try a |

|larger n. Let's try n = 41. |

|40 |26.50 |0.662 | |55.8 |1.40 |

| |

|Now we have 0.662 > 0.64 and 1.40 < 1.57. We need to go lower. As you continue this process, you will |

|eventually converge on a "best" solution. See the complete table on the next page: |

|35 |22.46 |0.642 | |49.8 |1.42 |

Thus a rough estimate of the required sample size is approximately 36 (n-1= 35).

-----------------------

df= n-1 = 4

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Ho is true

Ho is false

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