Statistics: Opinion Polls and Confidence Intervals
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Questions and Answers

What is the standard error of the sample proportion if 25% of a random sample of 200 people are opposed to a state lottery?

  • 6.1237
  • 0.0306 (correct)
  • 0.0353
  • 0.0094
  • 0.2500

    • Which condition for constructing a confidence interval for a proportion has the consultant failed to satisfy when estimating the proportion of shoppers making purchases?

  • np̂ ≥ 10
  • n ≥ 30
  • n(1-p̂ ) ≥ 10
  • The data is a random sample from the population of interest (correct)
  • The sample is less than 10% of the population

    • What does the phrase '95% confidence' mean in the context of a sample survey?

  • 95% of all Americans will spend between 9% and 15% more than what they spent last year.
  • There is a 95% chance that the percent who expect to spend more is between 9% and 15%.
  • We can be 95% confident that the method used to get the interval always gives the right answer.
  • 9% to 15% of all Americans will spend 95% of what they spent last year.
  • The method used to get the interval from 9% to 15%, when used over and over, produces intervals which include the true population percentage about 95% of the time. (correct)

    • What is the best way to correct for undercoverage in a telephone survey?

    <p>Throw this sample out and start over again with a better sampling method.</p> Signup and view all the answers

    What sample size is needed for a 95% confidence interval with a margin of error of no more than 3%?

    <p>1068</p> Signup and view all the answers

    Which of the following has the highest probability?

    <p>Randomly selecting a value less than 3 from a standard Normal distribution.</p> Signup and view all the answers

    How would the margin of error of a 99% confidence interval compare with a 95% interval using the same sample?

    <p>It would be larger, because higher confidence requires a larger margin of error.</p> Signup and view all the answers

    What is the appropriate t* value for a 98% confidence interval from a sample of n = 18?

    <p>2.567</p> Signup and view all the answers

    What is the standard error of x̅ for a mean height of a random sample of 25 males with a mean of 69.72 inches and a standard deviation of 4.15?

    <p>0.83</p> Signup and view all the answers

    For which of the samples would it be safe to construct a t-interval based on their distributions?

    <p>Sample Y only.</p> Signup and view all the answers

    What is the standard error of the estimate for a proportion of 0.20 from a sample of 900 individuals?

    <p>0.0133</p> Signup and view all the answers

    Which condition for constructing a confidence interval for a proportion has she failed to satisfy after counting cars with more than two occupants?

    <p>The data is an SRS from the population of interest.</p> Signup and view all the answers

    Which statement is a correct interpretation of a 95% confidence margin of error of 3% in a poll?

    <p>None of the above.</p> Signup and view all the answers

    What is the best way to correct for undercoverage in an email survey?

    <p>Throw this sample out and start over again with a better sampling method.</p> Signup and view all the answers

    How large should the sample size n be to achieve a 95% confidence interval for μ with a margin of error of ± 0.0001?

    <p>38,416</p> Signup and view all the answers

    Which of the following has the highest probability for selecting values between -2 and 2?

    <p>Randomly selecting a value between -2 and 2 from a standard Normal distribution.</p> Signup and view all the answers

    Which statement is correct regarding the width of a 99% confidence interval compared to a 95% interval?

    <p>The 99% interval is wider.</p> Signup and view all the answers

    What is the critical value t* for a t-distribution with 8 degrees of freedom that has a probability of 0.10 to the right?

    <p>1.397</p> Signup and view all the answers

    What is the standard error of the mean for the weights of 9 men with a mean of 175 pounds and a standard deviation of 15 pounds?

    <p>5</p> Signup and view all the answers

    For which of the samples would it be safe to construct a t-interval based on their distributions?

    <p>Sample Z only.</p> Signup and view all the answers

    Study Notes

    Opinion Polls and Confidence Intervals

    • Standard error of a sample proportion can be calculated; example: sample of 200 with 25% strongly opposed → 0.0306.
    • For estimating proportions, conditions must be met: np̂ ≥ 10 is essential, as seen in a mall survey with insufficient sample randomness.

    Confidence Level Interpretation

    • "95% confidence" means that if the method is repeated, the intervals constructed will capture the true population proportion 95% of the time.

    Sampling Bias and Data Representation

    • Undercoverage occurs if certain populations are excluded; remedy includes utilizing a better sampling method rather than changing confidence levels.
    • To mitigate bias in surveys (e.g., missing non-email users), starting over with an improved method is recommended.

    Determining Sample Size

    • To achieve a specified margin of error (e.g., ± 3%) at a 95% confidence level, specific sample sizes are necessary; for example, 1068 for estimating drug effectiveness in patients.

    Understanding Probability Distributions

    • Probabilities vary between distributions: higher for standard Normal than t-distributions, particularly when values are chosen from tails (e.g., less than 3).
    • With increasing confidence level (e.g., 95% to 99%), margin of error increases, thereby necessitating larger sample sizes.

    Confidence Interval Calculations

    • Critical values (t*) depend on degrees of freedom; to find the right critical value, refer to t-distribution tables for appropriate levels (e.g., 1.397 for 8 DF).

    Standard Error of the Mean

    • Standard error can be calculated using sample standard deviation and sample size; e.g., a mean weight of 175 pounds with a standard deviation of 15 pounds gives a standard error of 5.

    Sample Conditions for t-Intervals

    • For any confidence interval construction, it's crucial to check sample distribution robustness; plotting conditions help ascertain suitability.
    • Larger sample sizes with better distribution characteristics support safer t-interval calculations.

    Margin of Error Implications

    • Understanding that higher confidence levels necessitate wider confidence intervals emphasizes the trade-off between confidence and precision, crucial for making informed statistical analysis.

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    Quiz Team

    Description

    Explore the concepts of opinion polls, confidence intervals, and sampling bias in this quiz. Understand how to calculate the standard error of a sample proportion and interpret confidence levels. Learn about the importance of proper sampling methods to avoid bias and determine the appropriate sample size for accurate data representation.

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