POLI 205
Boomer18
Tokunaga2e_PPT_08.pdf
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Chapter 8
Estimating the Mean of a Population
Introduction to the Confidence Interval for the Mean
• So far, we have known what the population mean () was for our sample comparisons
• This is not the most realistic scenario
– Many times researchers do not know the population mean, but need to estimate it from the sample mean
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Introduction to the Confidence Interval for the Mean
• Example: A sample of 38 members of the Society of Industrial and Organizational Psychologists reported their annual salary
= 64,508.84
s = 14,523.92
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X
𝑋 = $64,508.84
Introduction to the Confidence Interval for the Mean
• Point estimate: single value used to estimate an unknown population parameter
• However, recall sampling error
– If we drew different samples, it’s unlikely that we would get the exact same mean again
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Introduction to the Confidence Interval for the Mean
• Confidence interval: range of values that, with a stated degree of confidence, may include the population mean
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The Confidence Interval for the Mean ( Not Known)
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The Confidence Interval for the Mean ( Not Known)
• State the desired level of confidence
– How confident do we want to be about the estimate of the parameter?
– The impracticalities of 100% confidence
– A solution: 95% confidence interval
o The researcher is 95% confident that the unknown lies within the interval
o .95 probability that the interval contains
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The Confidence Interval for the Mean ( Not Known)
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The Confidence Interval for the Mean ( Not Known)
• Calculate the confidence interval and confidence limits
– : point estimate (sample mean)
– t: critical value of the t-statistic for the desired level of confidence
– s : standard error of the mean
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X
X
The Confidence Interval for the Mean ( Not Known)
• Calculate the s : standard error of the mean
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The Confidence Interval for the Mean ( Not Known)
• Calculate the df and identify critical t value:
df = N – 1 = 38 – 1 = 37
• A 95% CI implies α = .05
– Using the table in the back of the book, locate critical value for α = .05 and df = 37
– t = 2.042
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The Confidence Interval for the Mean ( Not Known)
• Calculate the confidence interval:
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The Confidence Interval for the Mean ( Not Known)
• Calculate the confidence limits
• Upper limit:
= 64,508.84 + 4,811.14
= 69,319.98
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• Lower limit:
= 64,508.84 – 4,811.14
= 59,697.70
• 95% CI = (59,697.70, 69,319.98)
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The Confidence Interval for the Mean ( Not Known)
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• Draw conclusions about the confidence interval
– There is a .95 probability that the interval of $59,697.70 to $69,319.98 contains the mean income for the population of recent master’s degrees recipients in I-O psychology
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The Confidence Interval for the Mean ( Not Known)
Tokunaga, Fundamental Statistics for the Social Sciences, 2e SAGE Publishing, 2019
What Does the Confidence Interval Imply?
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What Does the Confidence Interval Imply?
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What Does the Confidence Interval Imply?
• The population mean does not change
– Samples, from which we develop our confidence interval, do change
– The probability statement regards the confidence interval, not the population mean
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The Confidence Interval for the Mean ( Not Known)
• This is an unlikely situation for a researcher to encounter
– Much of the process follows the procedures discussed earlier
• State the desired level of confidence
– Again, typically 95% confidence level
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The Confidence Interval for the Mean ( Known)
• Calculate the confidence interval and confidence limits
– : point estimate (sample mean)
– z: critical value of the z-statistic for the desired level of confidence
– : standard error of the mean
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X
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The Confidence Interval for the Mean ( known)
• Calculate the standard error of the mean
• Identify critical value of z – For 95% CI, the critical value of z =1.96
• Calculate the confidence interval
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The Confidence Interval for the Mean ( Known)
• Draw a conclusion about the confidence interval
• Explicitly identify
– The level of confidence
– The variable being measured
– The population whose mean is being estimated
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Factors Affecting the Size of the Confidence Interval for the Mean
• Sample size
– The larger the sample size, the narrower the CI
o Influence of sample size on standard error of the mean
– Narrower CI greater precision
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Factors Affecting the Size of the Confidence Interval for the Mean
• Sample size: We can also estimate the sample size needed for a particular width of a confidence interval
– N = necessary sample size – z = z-statistic associated with CI
– = estimated population standard deviation
– CI = desired width of the confidence interval Tokunaga, Fundamental Statistics for the Social Sciences, 2e
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Factors Affecting the Size of the Confidence Interval for the Mean
• Desired level of confidence: the higher the desired level of confidence, the wider the confidence interval
– A 99% CI is wider than the 95% CI
– A 90% CI is narrower than the 95% CI
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Factors Affecting the Size of the Confidence Interval for the Mean
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Factors Affecting the Size of the Confidence Interval for the Mean
• Choosing between different levels of confidence
– It’s a tradeoff: Would you rather be more certain that a given interval contains the population mean, or would you rather be more precise?
– Also depends on sample size
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Interval Estimation and Hypothesis Testing
• Concerns with hypothesis testing
– Interpreting statistical significance
o H0 and H1 may lead to dichotomized discussions of “significant” and “non-significant” results
o We may ignore non-significant findings, assuming that means there is no effect
o What about when p = .06? Is that the same as a result when p = .99?
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Interval Estimation and Hypothesis Testing
• Concerns with hypothesis testing
– Influence of sample size on statistical analyses
o Larger sample size leads to larger probability of rejecting the null hypothesis
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Interval Estimation and Hypothesis Testing
• A potential solution
– Rather than make a decision about significance, researchers could use the sample mean to develop a confidence interval for the mean in order to estimate the population mean
– Especially useful when the true population mean is unknown
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Looking Ahead
• This chapter
– Explored interval estimation and confidence intervals
• Next chapter
– We return to testing research hypotheses, specifically hypotheses regarding the difference between two means
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Summary
• Interval estimation
• Point estimate
• Confidence interval
• Confidence interval for the mean
• Confidence limits
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