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   Z-Test Formula

The Z-Test is implemented as follows:

Two Sample Test

Assumptions

The assumptions/conditions needed when testing the difference between two percentages (proportions) using Maritz Stats include:

  1. Measurement scale is at least nominal
  2. Random samples
  3. Weighting is not used
  4. The random variables are binomially distributed, meaning:
    Both random variables are defined in terms of the number of occurrences (e.g., number of successes, number of "Yes" responses, number of top box responses)
    There are a fixed number of trials (e.g., the number to be sampled is not determined by the number of successes)
    The result of each trial can be classified into one of two categories (e.g., success or failure, yes or no, top box or non-top box)
    The probability of success remains constant for each trial (e.g., the probability of randomly selecting a "success" does not change throughout the sample)
    Each trial of the experiment is independent of the other trials (e.g., the response by one respondent does not affect, and is not affected by, the response by another respondent)
  5. The sample sizes, taking into account the percents (proportions), are sufficiently large (i.e., each sample size times the corresponding percent of success is greater than 500 and each sample size times the corresponding percent of non-success is greater than 500, or, in terms of proportions, the sample size times the probability of success is greater than five, and the sample size times the probability of non-success is greater than 5 ( n1p1 > 5, n1(1 - p1) > 5, n2p2 > 5, and n2(1 - p2) > 5 )
  6. The hypothesis test is to determine if the difference between two population percents (proportions) is either equal to, less than, or greater than zero (as opposed to a non-zero constant)

Variables input:

n1 Number of valid responses for group 1
n2 Number of valid responses for group 2
y1 Number of responses to a particular selection for group 1
y2 Number of responses to a particular selection for group 2
p1 Percent of responses to a particular selection for group 1
p2 Percent of responses to a particular selection for group 2
N1 Population for group 1
N2 Population for group 2
n12 Number of overlapping respondents
r12 Correlation between group 1 and 2

If a finite population correction factor is used, compute for each group:

Compute the pooled variance estimate:

Compute standard error:

Compute the value of z:

Compute the result of the test:

The critical value is determined based on a table of z values, which determines the critical value based on the selected level of confidence. The computed z is compared to the critical value to determine if the difference is significant.

 

One Sample Test

Assumptions

The assumptions/conditions needed when testing one mean against an expected value using Maritz Stats include:

  1. Measurement scale is at least nominal
  2. Random sample
  3. Weighting is not used
  4. The random variable is binomially distributed, meaning:
    The random variable is defined in terms of the number of occurrences (e.g., number of successes, number of "Yes" responses, number of top-box responses)
    There are a fixed number of trials (e.g., the number to be sampled is not determined by the number of successes
    The result of each trial can be classified into one of two categories (e.g., success or failure, yes or no, top box or non-top box)
    The probability of success remains constant for each trial (e.g., the probability of randomly selecting a "success" does not change throughout the sample)
    Each trial of the experiment is independent of the other trials (e.g., the response by one respondent does not affect, or is affected by, the response by another respondent)
  5. The sample size, taking into account the percents (proportions), is sufficiently large (i.e., sample size times the percent of success is greater than 500 and the sample size times the percent of non-success is greater than 500, or, in terms of proportions, the sample size times the probability of success is greater than five (np > 5), and the sample size times the probability of non-success is greater than 5 (n(1 - p) > 5)

Variables input:

ps Sample proportion
p0 Hypothesized proportion
n Sample size
N Population

Compute the value of z:

Compute the result of the test:

The critical value is determined based on a table of z values, which determines the critical value based on the selected level of confidence. The computed z is compared to the critical value to determine if the difference is significant.