Statistics Exam Review

Understanding Key Concepts in Statistics

• Mean and Standard Deviation

  • The mean, denoted by ( \mu ), represents the average of a dataset.
  • Standard deviation, denoted by ( \sigma ), measures the spread of the dataset.

• Z-Score Calculation

  • The formula for the Z-score is given by:
    z = \frac{x - \mu}{\sigma}
  • Example: To calculate the Z-score for ( x = 132 ), ( \mu = 120 ), ( \sigma = 5 ):
    z = \frac{132 - 120}{5} = 2.4

• Power of a Test

  • The power of the test refers to the probability that it correctly rejects the null hypothesis.

Using Z-Score Tables

• For a Z-score of 2.4, referring to the Z-table gives a value of approximately 0.9918.
• Conversely, for a Z-score of 2.0, the Z-table yields a value of around 0.9772.

Understanding Sample Size and Probability

• Sample Size Impacts

  • A larger sample size produces a more accurate and reliable estimate of the mean.

• The formula for calculating the Z-score for a sample mean is:
  z = \frac{x - \mu}{\sigma / \sqrt{n}}

• Example: For ( n = 16 ), to calculate the probability of ( x > 124 ):

  • Compute the Z-score:
    z = \frac{124 - 120}{5 / \sqrt{16}} = 3.2

• Using the Z-table, a Z-score of 3.2 gives a principal value of approximately 0.9993, leading to a probability of:
  P(X > 124) = 1 - 0.9993 = 0.0007

Confidence Intervals

• The formula for a confidence interval is defined as:
  CI = \hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}
• Example: For a proportion with ( n = 348 ) and ( \hat{p} = 0.193 ):
  • Calculate ( z_{\alpha/2} ) for a 99% confidence level (approximately 2.575).
  • Then determine the confidence interval bounds.

Understanding Normal Approximation

• When moving from a binomial to a normal distribution, the continuity correction must be applied:

  • For example, when calculating probabilities for ( x < 25 ), consider ( x = 24.5 ) instead.

• Binomial Distribution to Normal

  • Mean for binomial is ( \mu = np ) and standard deviation is ( \sigma = \sqrt{np(1-p)} )
  • For a success count ( = 500 \times 0.5 ) and failure rate ( = 0.5 ):
    • Calculate mean and standard deviation. Then compute probabilities similarly as above.

Key Statistical Distributions & Concepts

• T-Distribution

  • The t-distribution is used for small sample sizes (typically ( n < 30 )).
  • The formula for the t-interval is:
    CI = \bar{x} \pm t_{\alpha/2} \frac{s}{\sqrt{n}}
  • An example would involve computing degrees of freedom: ( df = n - 1 ).

• Chi-Squared Tests

  • The chi-squared distribution is used in hypothesis testing.
  • Example involves finding critical values for chi-squared and calculating the intervals.

Final Exam Preparation Tips

• Review all sample problems similar to those provided in class.
• Make sure you understand each statistical concept and its applications.
• Practice using the calculator for statistical tests to be proficient during the exam.
• Bring equations or charts you find helpful, including Z-tables, and review the material thoroughly before the final exam.