PSCI 2702 Chapter 6
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What is the significance of a sample mean being within ±1.96 Z’s from the mean of the sampling distribution?

  • It guarantees the population mean is included.
  • It suggests the population mean is likely included. (correct)
  • It shows the sample mean is reliable.
  • It indicates that the sample is too small.

    • What sample size is considered large enough for the Central Limit Theorem to apply?

  • 75 or more cases
  • 30 or more cases
  • 50 or more cases
  • 100 or more cases (correct)

    • What condition must be met to use the standardized normal distribution with small samples?

  • The population from which the sample is taken must be normally distributed. (correct)
  • The sample must have more than 50 cases.
  • The sample must be randomly selected.
  • The population must have a sample size less than 30.

    • What trend was observed in Canadian perceptions of crime from 1993 to 2014?

    <p>Perceptions of crime generally declined.</p> Signup and view all the answers

    In what year did the proportion of Canadians who felt unsafe walking alone after dark drop to 6.2%?

    <p>2014</p> Signup and view all the answers

    What happens if a sample mean is more than ±1.96 Z’s away from the mean of the sampling distribution?

    <p>The population mean may not be included in the interval.</p> Signup and view all the answers

    What was the percentage of respondents who believed their neighborhood had above-average crime in 1993?

    <p>11.6%</p> Signup and view all the answers

    Why is careful consideration of sample size important in statistical analysis?

    <p>Larger samples lead to better estimates of population parameters.</p> Signup and view all the answers

    What must be implicitly assumed while using the Z distribution for samples smaller than 100 cases?

    <p>The population must be normally distributed.</p> Signup and view all the answers

    Which of the following years had the lowest reported concern regarding feeling unsafe after dark?

    <p>2014</p> Signup and view all the answers

    Which confidence interval represents the percentage of Canadians who felt unsafe walking alone after dark in 2014?

    <p>5.66% to 6.74%</p> Signup and view all the answers

    What is the primary reason for calculating confidence intervals in statistics?

    <p>To determine the reliability of the sample statistic.</p> Signup and view all the answers

    What is the first step in constructing an error bar graph for a confidence interval of a sample mean?

    <p>Plot the sample mean with a symbol on the graph.</p> Signup and view all the answers

    When invoking the Central Limit Theorem, what is a key outcome regarding the shape of the sampling distribution?

    <p>It becomes approximately normal for large sample sizes.</p> Signup and view all the answers

    What was the sample size in the study conducted in 1999?

    <p>25,876</p> Signup and view all the answers

    In the context of the error bar graph, what does the area bounded by the two error bars represent?

    <p>The width of the confidence interval.</p> Signup and view all the answers

    How much did the proportion of respondents feeling unsafe after dark change from 1993 to 2014?

    <p>Decreased by 21.1%</p> Signup and view all the answers

    Given a sample mean of 105 and a standard deviation of 15, what are the lower and upper limits of the 95% confidence interval calculated using the error bar method?

    <p>99.31 and 110.69.</p> Signup and view all the answers

    During the construction of a confidence interval, which of the following steps involves the sample standard deviation?

    <p>Dividing the standard deviation by the square root of sample size.</p> Signup and view all the answers

    Which year reported a 3.9% belief that neighborhoods had higher crime?

    <p>2014</p> Signup and view all the answers

    What is the purpose of the small horizontal line at the ends of the error bars in a graph?

    <p>To mark the limits of the confidence interval.</p> Signup and view all the answers

    Which of the following formulas should be used to construct the confidence interval for a sample mean?

    <p>Confidence Interval = sample mean ± t * (s/n)</p> Signup and view all the answers

    What represents the sample mean in the graph of the error bar for the confidence interval?

    <p>A dot or symbol.</p> Signup and view all the answers

    How can you determine the t score required for constructing a confidence interval?

    <p>By selecting the alpha level and the degrees of freedom.</p> Signup and view all the answers

    What is the formula for a confidence interval when the population standard deviation is known and the sample size is large?

    <p>c.i. = X ± Z(σ/√n)</p> Signup and view all the answers

    Which formula would you use to calculate the required sample size for estimating a population proportion when the margin of error (ME) is known?

    <p>n = Z² × Pμ(1−Pμ)/ME²</p> Signup and view all the answers

    What is the purpose of a confidence interval?

    <p>To estimate the range in which a population parameter likely falls</p> Signup and view all the answers

    If the sample mean is 5.2, the population standard deviation is 0.7, and the sample size is 157, what is the z-value for constructing a 95% confidence interval?

    <p>1.96</p> Signup and view all the answers

    In which situation would you prefer to use the t-distribution for a confidence interval?

    <p>When population standard deviation is unknown and sample size is small</p> Signup and view all the answers

    What does the margin of error in a confidence interval indicate?

    <p>The maximum sampling error that can occur</p> Signup and view all the answers

    Which statement is true regarding confidence intervals and sample size?

    <p>A smaller sample size decreases the precision of the confidence interval</p> Signup and view all the answers

    If you have a population proportion of 0.5 and a desired margin of error of 0.05, which sample size would be suitable?

    <p>n = (Z² * 0.5 * (1 - 0.5)) / 0.05²</p> Signup and view all the answers

    What is the mean number of cigarettes smoked per day by daily smokers in Canada?

    <p>14.98</p> Signup and view all the answers

    What is the estimated 95% confidence interval for the mean number of cigarettes smoked by daily smokers in Canada?

    <p>13.76 to 16.19</p> Signup and view all the answers

    How many respondents indicated they had a seasonal flu shot according to the output?

    <p>1046</p> Signup and view all the answers

    What percentage of respondents reported not having a seasonal flu shot?

    <p>39.5%</p> Signup and view all the answers

    What could be the first step to calculate the sample proportion in SPSS?

    <p>Use the Frequencies command.</p> Signup and view all the answers

    Which of the following statements is true about the sample statistics used for constructing confidence intervals?

    <p>They are essential to estimate confidence intervals.</p> Signup and view all the answers

    What is the total number of respondents in the sample regarding the seasonal flu shot?

    <p>1730</p> Signup and view all the answers

    In constructing a confidence interval for a sample proportion, what needs to be done with the percentages?

    <p>Convert them to decimals.</p> Signup and view all the answers

    Study Notes

    Confidence Interval for Sample Means

    • 95% of all possible sample means fall within ±1.96 Z's (or ±2.08 IQ units) of the mean of the sampling distribution
    • Larger samples (100 or more cases) allow the Central Limit Theorem to be applied
    • For smaller samples (<100 cases), the population needs to be normally distributed to use the standardized normal (Z) distribution for confidence intervals
    • If the sample size is large OR the population is normally distributed, the sampling distribution is considered normal in shape

    Graphing a Confidence Interval

    • Error bars graph is used to depict a confidence interval of a sample mean
    • The sample mean is plotted as a dot in the center of the graph
    • A vertical line (error bar) is drawn from the sample mean to the lower and upper limits of the confidence interval
    • The area bounded by the error bars represents the width of the confidence interval

    Constructing Confidence Intervals

    • Confidence intervals are calculated using formulas that take different factors into account, depending on the know information
    • For sample means with known population standard deviation and a large sample size OR a normally distributed population:
      • Confidence interval = Sample mean ± Z * (Population standard deviation / Square root of sample size)
    • For sample means with unknown population standard deviation and a large sample size OR a normally distributed population:
      • Confidence interval = Sample mean ± t * (Sample standard deviation / Square root of (sample size - 1))
    • For sample proportions with known population proportion and a large sample size:
      • Confidence interval = Sample proportion ± Z * (Square root of (Population proportion * (1 - Population proportion) / Sample size))
    • For sample proportions with unknown population proportion and a large sample size:
      • Confidence interval = Sample proportion ± Z * (Square root of (Sample proportion * (1 - Sample proportion) / Sample size))
    • For required sample size for a mean:
      • Sample size = (Z^2 * population standard deviation^2) / margin of error^2
    • For required sample size for a proportion:
      • Sample size = (Z^2 * population proportion * (1 - population proportion)) / margin of error^2

    Example: Crime Perception in Canada

    • Canadian study investigated perceptions of crime in neighborhoods from 1993 to 2014
    • Data showed a decrease in the proportion of Canadians who felt unsafe walking in their neighborhood after dark
    • The perception of crime being higher in the neighbourhood also decreased over the same period
    • 95% confidence intervals for proportions were calculated using a formula
    • The results showed that in 2014, between 5.66% and 6.74% of Canadians felt unsafe walking alone after dark
    • The conclusion: Canadians’ perception of crime generally declined over 21 years

    Student Resources

    • Companion website for the 5th Canadian edition of Statistics: A Tool for Social Research and Data Analysis
    • Website address: www.cengage.com/healey5ce

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    Related Documents

    Statistics: Estimation Procedures (Chapter 6)

    Description

    This quiz covers the essential concepts of confidence intervals for sample means, including the application of the Central Limit Theorem and methods for graphing confidence intervals. Understand how to construct confidence intervals and interpret error bars effectively in graphical representations.

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