POLI 205
Boomer18
Tokunaga2e_PPT_07.pdf
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Chapter 7
Testing One Sample Mean
A Brief Review…
• Descriptive statistics allow us to understand the shape of a frequency distribution of numbers
• Inferential statistics allow us to evaluate how different a sample mean is from the population
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The Sampling Distribution of the Mean
• A distribution of values of the sample mean for an infinite number of samples of size N that are randomly drawn from the population
• Normal distribution consists of individual scores
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The Sampling Distribution of the Mean
• Imagine that we know that the age of people completing a PhD is normally distributed with = 30 and = 7
– If we draw several samples of size N = 3
o Will the samples have the same average? o Will the samples be the same as each other?
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The Sampling Distribution of the Mean
• If we continue to draw samples of size N = 3, and use the mean of each sample as a data point, we can generate a frequency distribution that consists of means
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Characteristics of the Sampling Distribution of the Mean
• Modality – The mean of the new
distribution is • Symmetry
– Approximately normal • Variability
– Standard error of the mean: average deviation of the sample mean from the population mean
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The Sampling Distribution of the Mean
• Central limit theorem
– Sample means are approximately normally distributed
o Mean = µ o Standard deviation = standard error of the mean o When samples are large enough (N ≥ 30)
Tokunaga, Fundamental Statistics for the Social Sciences, 2e SAGE Publishing, 2019
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An Example
• A new reading program tests students’ word count per minute (WCPM)
• It is known that in the normal population, children can read with µ = 124.81 WCPM, and = 43.26
• A sample of 20 fourth-grade children in the new program can read = 150.35
• Does this new reading program change reading ability?
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Inferential Statistics: Testing One Sample Mean ( Known)
• State H0 and H1:
H0: µ = 124.81
H1: µ ≠ 124.81
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Inferential Statistics: Testing One Sample Mean ( Known)
• Make a decision about H0 – Set , identify critical values, and state a decision rule – = .05 (two-tailed) o Because it is a non-directional hypothesis, the 5%
region of rejection is split into two regions, one in each tail
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Inferential Statistics: Testing One Sample Mean ( Known)
• Make a decision about H0 – Rejection rule: If z < –1.96 or > 1.96, reject H0;
otherwise, do not reject H0
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Inferential Statistics: Testing One Sample Mean ( Known)
• Make a decision about H0 – Calculate a statistic: z-test for one mean
– Because we’re now using means, the formula is modified:
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Inferential Statistics: Testing One Sample Mean ( Known)
• Make a decision about H0 • Population standard error of the mean
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Inferential Statistics: Testing One Sample Mean ( Known)
• Make a decision about H0 – Calculate a statistic: z-test for one mean
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Inferential Statistics: Testing One Sample Mean ( Known)
• Make a decision about H0 – Make a decision whether to reject the null hypothesis
o 2.64 > 1.96, therefore reject H0 (p < .05) o 2.64 > 2.58, therefore reject H0 (p < .01)
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Inferential Statistics: Testing One Sample Mean ( Known)
• Draw a conclusion
– The number of words correct per minute (WCPM) (M = 150.35) in a sample of 20 fourth-grade students was significantly greater than the national normative sample (μ = 124.81), z = 2.64, p < .01
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Inferential Statistics: Testing One Sample Mean ( Known)
• Relate the result of the analysis to the research hypothesis
– “This study revealed that teaching explicit, systematic reading comprehension strategies to fourth graders is likely to increase reading comprehension skills”
(Reed et al, 2007, p. 64)
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Inferential Statistics: Testing One Sample Mean ( Known)
• Assumptions of the z-test for one mean
– Random sampling
– Interval or ratio data
– Assumption of normality
o Although the z-test is robust
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Inferential Statistics: Testing One Sample Mean ( Known)
• A sample of 15 students scored an average of 23 points on a quiz
• Based on past administrations of the quiz, the professor knew that scores were normally distributed, with = 20 and = 5
• Is this new sample of students scoring significantly better than the population?
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Introduction to the t Distribution
• Student t-distribution
– Distribution of values of the t-statistic
– t-statistics are used to estimate the population
o When is unknown o Uses the properties of the sample to estimate the
population
– Shares many characteristics with the standard normal distribution
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Introduction to the t Distribution
• Modality
– The mean of the t-distribution is 0
• Symmetry
– Approximately normal, but changes shape with different sample sizes
• Variability
– Standard error of the mean: average deviation of the sample mean from the estimated population mean
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Introduction to the t Distribution
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An Example
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• It is known than in the U.S., people typically die at age 75
• A sample of 17 students were asked to predict when they would die
• Their scores averaged = 84.00, with s = 8.01
• Do students believe that they will live longer than the national average?
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Inferential Statistics: Testing One Sample Mean ( Not Known)
• State H0 and H1:
H0: µ = 75.00
H1: µ ≠ 75.00
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Inferential Statistics: Testing One Sample Mean ( Not Known)
• Make a decision about H0 – Calculate the df:
o The number of values that are free to vary when using a sample statistic to estimate a population parameter
df = N – 1
df = 17 – 1 = 16
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Inferential Statistics: Testing One Sample Mean ( Not Known)
• Make a decision about H0 – Set , identify critical values, and state a decision rule – = .05 (two-tailed)
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Inferential Statistics: Testing One Sample Mean ( Not Known)
• Make a decision about H0 – Rejection rule: If t < –2.120 or > 2.120, reject H0;
otherwise, do not reject H0
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Inferential Statistics: Testing One Sample Mean ( Not Known)
• Make a decision about H0 – Calculate a t-statistics:
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Inferential Statistics: Testing One Sample Mean ( Not Known)
• Make a decision about H0 • Estimated population standard error of the mean:
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Inferential Statistics: Testing One Sample Mean ( Not Known)
• Make a decision about H0 • Calculate a statistic: t-statistics
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Inferential Statistics: Testing One Sample Mean ( Not Known)
• Make a decision about H0 – Make a decision whether to reject the null hypothesis
o t = 4.63> 2.120, therefore reject H0 (p < .05) o t = 4.63 > 2.921, therefore reject H0 (p < .01)
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Inferential Statistics: Testing One Sample Mean ( Not Known)
• Draw a conclusion
– The average estimated age of death (M = 84.00 years) for the 17 class members in this sample was significantly greater than the actual population average of 75 years, t(16) = 4.63, p < .01
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Inferential Statistics: Testing One Sample Mean ( Not Known)
• Relate the result of the analysis to the research hypothesis
– The result of this analysis supports the research hypothesis that people will provide estimates of age of death greater than the average life expectancy in the population
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Inferential Statistics: Testing One Sample Mean ( Not Known)
• Assumptions of the t-test
– Random sampling
– Interval or ratio data
– Assumption of normality
o Although the t-test is robust
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Factors Influencing the Decision about the Null Hypothesis
• Sample size
– The larger the sample size, the greater the probability of rejecting the null hypothesis
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N = 20 N = 10
t = 3.19 t = 2.25
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Factors Influencing the Decision about the Null Hypothesis
• Alpha: the larger the value of , the greater the likelihood of rejecting the null hypothesis
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Factors Influencing the Decision about the Null Hypothesis
• Directionality of the alternative hypothesis
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Looking Ahead
• This chapter
– Used the z-test and t-test to test the difference between a sample mean and a hypothesized population mean
• Later chapters
– Will examine procedures designed to test hypotheses in different research situations
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