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Discussion Notes

t-Test for Two Independent Samples

1. This test is based on the t distribution, a set of values based on the normal curve that is suitable for small samples ( (2 ( There is a significant difference between the mean…/The mean of group 1 is less than…

2. State the level of significance, (

a. Significant difference: ( = 0.05

b. Highly significant difference ( = 0.01

3. Follow Steps 4 and 5 in the manual to get tcalculated. N1 + N2 – 2 is known as df or degrees of freedom.

4. Use Critical values of t table to get tcritical.

5. Compare tcalculated and tcritical and conclude.

a. If tcalculated < tcritical, then accept H0 and reject HA: There is no significant difference…

b. If tcalculated > tcritical, then reject H0 and accept HA: There is a significant difference…

|20 yr |60 yr olds |

|olds | |

|27 |26 |

|26 |29 |

|21 |29 |

|24 |29 |

|15 |27 |

|18 |16 |

|17 |20 |

|12 |27 |

|13 | |

Exercise:

The scores on a vocabulary test of a group of 20 year olds and a group of 60 year olds are shown on the right. At the 0.05 level of significance, is there a difference between the scores on a vocabulary test of a population of 20-year-olds and a population of 60-year olds? Perform the appropriate t test to find out.

Discussion Notes

t-Test for Two Correlated Samples

1. This test is based on the t distribution, a set of values based on the normal curve that is suitable for small samples ( (2 ( There is a significant difference between the mean…/The mean of group 1 is less than…

2. State the level of significance, (

a. Significant difference: ( = 0.05

b. Highly significant difference ( = 0.01

3. Follow Steps 4 and 5 in the manual to get tcalculated. N – 1 is known as df or degrees of freedom.

4. Use Critical values of t table to get tcritical.

5. Compare tcalculated and tcritical and conclude.

a. If tcalculated < tcritical, then accept H0 and reject HA: There is no significant difference…

b. If tcalculated > tcritical, then reject H0 and accept HA: There is a significant difference…

Exercise:

|water |alcohol |

|16 |13 |

|15 |13 |

|11 |10 |

|20 |18 |

|19 |17 |

|14 |11 |

|13 |10 |

|15 |15 |

|14 |11 |

|16 |16 |

An experiment is conducted on the effect of alcohol on perceptual motor ability. Ten subjects are each tested twice, once after having two drinks and once after having two glasses of water. The two tests were given on two different days to give the alcohol a chance to wear off. Half of the subjects were given alcohol first and half were given water first. The scores of the 10 subjects are shown on the right. The first number for each subject is their performance in the "water" condition. Higher scores reflect better performance.

At the 0.05 level of significance, does the alcohol have a significant effect on perceptual motor ability? Perform the appropriate t test to find out.

Discussion Notes

Single Factor Analysis of Variance (ANOVA)

1. Compares the variation and means between three or more groups due to only one experimental variable.

2. Groups need not have the same number of samples.

3. There has to be more than 1 replicate per treatment.

4. Variables measured need to be either interval or ratio.

5. Tests if there is a significant difference between the population means of three or more groups.

Steps:

1. Describe groups being compared and state null hypothesis and alternative hypotheses.

a. H0 : (1 = (2 = (3( There is no significant difference between the mean…

b. HA : (1 ( (2 ( (3 ( There is a significant difference between the mean…

2. State the level of significance, (

a. Significant difference: ( = 0.05

b. Highly significant difference ( = 0.01

3. Follow Steps 2 to 4 in the manual to get fcalculated. t is the total number of groups while r is the number of replicates per treatment.

4. Use f statistic table to get fcritical.

5. Compare fcalculated and fcritical and conclude.

a. If fcalculated < fcritical, then accept H0 and reject HA: There is no significant difference…

b. If fcalculated > fcritical, then reject H0 and accept HA: There is a significant difference…

Exercise:

1. It is suspected that the higher priced cars are assembled with greater care than lower-priced cars. To investigate if this claim has any basis, a large luxury model, A, a medium-sized sedan B, and a subcompact economy car C were compared for defects when they arrived at the dealer's showroom. All cars were manufactured by the same company. The number of defects for three of each of the models are shown below. Test the hypothesis that the average number of defects is the same for the 3 models (at the 0.05 level of significance).

2. In a biological experiment, 3 concentrations of a certain hormone are used to enhance the growth of a certain type of plant over a period of 1 month. The following growth data in mm were recorded for the plants that survived.

Table 2. Plant growth (mm)

Concentration of hormone

10% 5% 0%

82 77 68

87 84 73

94 86 69

Is there a significant difference in the average growth of these plants for the different concentrations of the hormone? Use a 0.05 level of significance.

Discussion Notes

Chi square test

1. This is a goodness-of-fit test is used when data is of the nominal type.

2. Two or more independent groups are being compared.

3. Tests if there is a significant difference between observed and expected frequencies.

Steps:

1. Describe groups being compared and state null hypothesis and alternative hypotheses.

a. H0 : There is no significant difference between the preferences of…

b. HA : There is a significant difference between the preferences of…

2. State the level of significance, (

a. Significant difference: ( = 0.05

b. Highly significant difference ( = 0.01

3. Follow Steps 3 to 8 in the manual to get tcalculated. N– 1 is known as df or degrees of freedom.

4. Use Critical values of (2 table to get (2critical.

5. Compare (2calculated and (2critical and conclude.

a. If (2calculated < (2critical, then accept H0 and reject HA: There is no significant difference…

b. If (2calculated > (2critical, then reject H0 and accept HA: There is a significant difference…

The following data are real. The cumulative number of AIDS cases reported for Santa Clara County through December 31, 2003, is broken down by ethnicity as follows:

Ethnicity Number of Cases

White 2032

Hispanic 897

African-American 372

Asian, Pacific Islander 168

Native American 20

Total = 3489

The percentage of each ethnic group in Santa Clara County is as follows:

Ethnicity Percentage of total county population Number expected (round to 2 decimal places)

White 47.79% 1667.39

Hispanic 24.15%

African-American 3.55%

Asian, Pacific Islander 24.21%

Native American 0.29%

Total = 100%

Expected Results

If the ethnicity of AIDS victims followed the ethnicity of the total county population, fill in the expected number of cases per ethnic group.

Goodness-of-Fit Test

Perform a goodness-of-fit test to determine whether the make-up of AIDS cases follows the ethnicity of the general population of Santa Clara County.

Discussion Notes

Friedmann’s test

1. This test is used when data is of the ordinal type.

2. Three or more matched groups are being compared.

3. Tests if groups have come from the same population.

Steps:

1. Describe groups being compared and state null hypothesis and alternative hypotheses.

a. H0 : The samples come from the same population. There is no significant difference between the acceptability of the samples of…

b. HA : The samples come from populations with different distributions. There is a significant difference acceptability of the samples of…

2. State the level of significance, (

a. Significant difference: ( = 0.05

b. Highly significant difference ( = 0.01

3. Follow Steps 4 and 5 in the manual to get Scalculated. m is the number of blocks and n is the number of groups being compared.

4. Use Critical values of S table to get Scritical.

5. Compare Scalculated and Scritical and conclude.

a. If Scalculated < Scritical, then accept H0 and reject HA: There is no significant difference…

b. If Scalculated > Scritical, then reject H0 and accept HA: There is a significant difference…

Exercise:

Five judges were asked to rate the Paskorus Presentations of three sections. The results are as follows. Use the Friedman’s test to determine if there is a significant difference in the acceptability of the three sections’ presentations.

|Judges |Section 1 |Section 2 |Section 3 |

|1 |7 |7 |8 |

|2 |6 |9 |8 |

|3 |4 |6 |5 |

|4 |6 |5 |8 |

|5 |7 |9 |9 |

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