Statistics Exam Review

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.

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.

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

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.

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.

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.

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.