Math 219: Section 7.2-10.3.1

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lazvertiigo
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Math 219: Section 7.2-10.3.1
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2014-11-14 13:20:54
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SCC Math219 Stewart
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SCC Statistics and Prob., Stewart
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  1. For Z = [X-μ]/σ, the random variable Z is a _______________.
    Standard Normal Distribution
  2. There are several ways to calculate the area under the standard normal curve.  The three different area calculations are...
    area to the left, to the right, and in between the z-score(s).
  3. Many of the statistical tests that we perform on small data sets (sample size less than ___) require that the population from which the sample is drawn be _______________.
    Many of the statistical tests that we perform on small data sets (sample size less than 30) require that the population from which the sample is drawn be normally distributed.
  4. We have said that a random variable X is normally distributed, or at least approximately normal, provided the histogram of the data is ______ and__________
    we have said that a random variable X is normally distributed, or at least approximately normal, provided the histogram of the data is symmetric and bell-shaped
  5. What defines a graph that plots observed data versus normal scores.
    A normal probability plot is a graph that plots observed data versus normal scores.
  6. A normal probability plot is a graph that plots observed data versus __________.
    A normal probability plot is a graph that plots observed data versus normal scores.
  7. What defines the expected z-score of the data value, assuming that the distribution of the random variable is normal.
    A normal score is the expected z-score of the data value, assuming that the distribution of the random variable is normal.
  8. The expected Z-score of an observed value will depend upon the number of _________ in the data set.
    The expected Z-score of an observed value will depend upon the number of observations in the data set.
  9. The idea behind finding the expected z-score is that, if the data comes from normally distributed population, we could predict _______ _________ _______.
    The idea behind finding the expected z-score is that, if the data comes from normally distributed population, we could predict the area to the left of each of the data value.
  10. T/F:
    If sample data is taken from a population that is normally distributed, a normal probability plot of the actual values versus the expected Z-scores will be approximately linear.
    True
  11. A new sample mean can be calculate each time a new sample is taken. In this way, the sample mean can be analyzed as _____________.
    A new sample mean can be calculate each time a new sample is taken. In this way, the sample mean can be analyzed as a random variable.
  12. Being able to approximately calculate the distribution of the sample mean is a critical tool for _________.
    Being able to approximately calculate the distribution of the sample mean is a critical tool for inference.
  13. Describe the sampling distribution of the sample mean.
    Because the sample mean is a random variable, the sample mean has a mean, and standard deviation, and probability distribution. This is called the sampling distribution of the sample mean.
  14. Because the sample mean is a random variable, the sample mean has three things; what are they?
    Because the sample mean is a random variable, the sample mean has a mean, and standard deviation, and probability distribution.
  15. T/F:
    If we know that the population has a normal distribution, then the sampling distribution will not be normal.
    • False:
    • If we know that the population has a normal distribution, then the sampling distribution will also be normal.
  16. T/F:

    If we know that the population has a normal distribution then the sampling distribution will be normally distributed, have a mean equal to the mean of the population, and have a standard deviation less than the standard deviation of the population.
    True
  17. The standard deviation of the sampling distribution of 𝑥̅ is called the _______ _______and is denoted σ𝑥̅.
    standard error of the mean
  18. T/F:
    If a random variable X is normally distributed, the distribution of the sample mean σ𝑥̅ is normally distributed.
    • False:
    • If a random variable X is normally distributed, the distribution of the sample mean 𝑥̅ is normally distributed.
  19. What is the Central Limit Theorem (conceptual)?
    Regardless of the shape of the distribution of a population, the sampling distribution of 𝑥̅ is approximately normal as the sample size n increases.
  20. T/F:
    The rule of thumb is if n≥10, this is a good approximation.
    The rule of thumb is if n≥30, this is a good approximation.
  21. What defines the process of using sample data to estimate the value of a population parameter?
    Estimation is the process of using sample data to estimate the value of a population parameter.
  22. What defines the value of a statistic that estimates the value of a parameter.
    A point estimate is the value of a statistic that estimates the value of a parameter.
  23. A _____ _______ for an unknown parameter consist of an interval of numbers.
    A confidence interval for an unknown parameter consist of an interval of numbers.
  24. The ____ ______ represents the expected proportion of intervals that will contain the parameter if a large number of different samples is obtained.
    The level of confidence represents the expected proportion of intervals that will contain the parameter if a large number of different samples is obtained.
  25. T/F:
    Confidence interval estimates for the population proportion are of the form
    Point estimate ± population proportion
    • False:
    • Confidence interval estimates for the population proportion are of the form
    • Point estimate ± margin of error
  26. The margin of error of a confidence interval estimate of a parameter depends on three factors: What are they?
    The margin of error of a confidence interval estimate of a parameter depends on three factors: level of confidence, sample size, and standard deviation of the population.
  27. T/F:
    For Sample Size: As the size of the random sample increases, the margin of error decreases.
    True
  28. T/F:
    For the standard deviation of the population: The more spread there is in the population, the smaller the interval will be for a given confidence level.
    • False:
    • The more spread there is in the population, the wider the interval will be for a given confidence level.
  29. T/F:
    For level of confidence: as the level of confidence increases, the margin of error increases.
    True
  30. Interpret the Confidence Interval.
    A [(1 − α) ∙ 100%] confidence interval indicates that, if we obtain many simple random samples of size n from the population whose parameter is unknown, then [(1 − α) ∙ 100%] of the intervals will contain the parameter.
  31. T/F:
    The number of degrees of freedom, n−1, is crucial for the t-distribution since this depends on the population proportion size.
    • False:
    • The number of degrees of freedom, n−1, is crucial for the t-distribution since this depends on the sample size.
  32. T/F:
    Properties of the t-Distribution
    The t-distribution is the same for different degrees of freedom.
    • False:
    • The t-distribution is different for different degrees of freedom.
  33. T/F:
    Properties of the t-Distribution

    The t-distribution is centered at 0 and is symmetric about 0.
    True
  34. T/F:
    Properties of the t-Distribution

    The area under the curve is 0.5.
    • False:
    • The area under the curve is 1.
  35. T/F:
    Properties of the t-Distribution
    As t increases or decreases without bound, the graph approaches, but never equals, zero.
    True
  36. T/F:
    Properties of the t-Distribution

    The area in the tails of the t-distribution is smaller than the area in the tails of the standard normal distribution, because we are using s as an estimate of σ, thereby introducing further variability into the t- statistic.
    • False:
    • The area in the tails of the t-distribution is a little greater than the area in the tails of the standard normal distribution, because we are using s as an estimate of σ, thereby introducing further variability into the t- statistic.
  37. T/F:
    Properties of the t-Distribution

    As the sample size n increases, the density curve of t gets closer to the standard normal density curve.
    True.
  38. Properties of the t-Distribution:

    As the sample size n increases, the density curve of t gets closer to the standard normal density curve.  This result occurs because, as the sample size n increases, the values of s get closer to the values of σ, by the Law of ____________.
    This result occurs because, as the sample size n increases, the values of s get closer to the values of σ, by the Law of Large Numbers.
  39. T/F:
    When constructing a [(1−α)∙100%] Confidence Interval for μ with unknown σ, the interval is exact when the population is normally distributed, but approximately correct for non-normal population, where n is large enough.
    True.
  40. T/F:
    Hypothesis testing and estimation are similar approaches to two similar problems.
    • False:
    • Hypothesis testing and estimation are two different approaches to two similar problems.
  41. Hypothesis testing and estimation are part of _________.
    Inferential Statistics.
  42. What defines a statement or claim regarding a characteristic of one or more populations?
    A hypothesis is a statement or claim regarding a characteristic of one or more populations.
  43. What defines the procedure, based on sample evidence and probability used to test statements regarding a characteristic of one or more populations?
    Hypothesis testing is a procedure, based on sample evidence and probability used to test statements regarding a characteristic of one or more populations.
  44. T/F:
    If population data are available, there is no need for inferential statistics.
    True.
  45. What are the steps in Hypothesis Testing? (3 steps)
    Step 1. A statement is made regarding the nature of the population.

    Step 2. Sample data is collected to test the statement.

    Step 3. The data are analyzed to assess the plausibility of the statement.
  46. Since claims can be either true or false, hypothesis testing is based on two types of hypothesis: ______ and ______.
    Since claims can be either true or false, hypothesis testing is based on two types of hypothesis: null and alternative.
  47. What defines the statement to be tested. We denote this by H0?
    The null hypothesis is the statement to be tested. We denote this by H0.
  48. What defines the claim to be tested. We denote this by H1?
    The alternative hypothesis is the claim to be tested. We denote this by H1.
  49. What are the different types of null hypothesis and alternative hypothesis pairs?
    Two-tailed, Left-tailed, and Right-tailed.
  50. T/F:
    Two-tailed test: test whether the parameter is either equal to, versus not equal to, some random variable.
    • False:
    • Two-tailed test: test whether the parameter is either equal to, versus not equal to, some value.
  51. T/F:
    Left-tailed test: test whether the parameter is either equal to, versus less than, some value.
    True.
  52. T/F:
    Right-tailed test: test whether the parameter is either equal to, versus greater than, some value.
    True.
  53. Define the type of error:
    If we reject stating the null hypothesis is false, but the null is true.
    Type I error.
  54. Define the type of error:
    If we do not reject, stating null hypothesis could be true, but the null hypothesis is actually false.
    Type II error.
  55. T/F:
    The level of significance, α, is the probability of making a Type II error.
    • False:
    • The level of significance, α, is the probability of making a Type I error.
  56. The probability of making a Type II error is represented by ___.
    β
  57. T/F:
    As the probability of Type I error increases, the probability of a Type II error decreases, and vice-versa.
    True.
  58. When observed results are unlikely under the assumption that the null hypothesis is true, we say the result is _____ _____.
    When observed results are unlikely under the assumption that the null hypothesis is true, we say the result is statistically significant.
  59. T/F:
    When results are found to be statistically significant, we accept the null hypothesis.
    • False:
    • When results are found to be statistically significant, we reject the null hypothesis.
  60. What are the three equivalent ways to perform a hypothesis test that reach the same conclusion?
    The methods are the classical approach, P-value approach, or confidence interval approach.
  61. T/F:
    Classical Approach: If the sample proportion is too many standard deviations from the proportion stated in the null hypothesis, we accept the null hypothesis.
    • False:
    • Classical Approach: If the sample proportion is too many standard deviations from the proportion stated in the null hypothesis, we reject the null hypothesis.
  62. P-value approach: If the sample proportion as extreme or more extreme than the one obtained is _____ under the assumption the statement in the null hypothesis is true, reject the null hypothesis.
    P-value approach: If the sample proportion as extreme or more extreme than the one obtained is small under the assumption the statement in the null hypothesis is true, reject the null hypothesis.
  63. What are the initial conditions for Testing Hypothesis Regarding a Population Proportion, p?
    • The sample is obtained by simple random sampling

    • np0(1 – p0) ≥ 10

    • The sampled values are independent of each other.
  64. What are the initial conditions for Testing Hypotheses Regarding a Population Mean, μ?
    • The sample is obtained using simple random sampling.

    • The sample has no outliers, and the population from which the sample is drawn is normally distributed or the sample size is large (n ≥ 30).

    • The sampled values are independent of each other.
  65. What do we call a procedure with minor departures from normality that will not adversely affect the results of the test?
    The procedure is robust.

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