One-sample Hypothesis Test for p

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Large Sample Proportion Hypothesis Testing
14:31 — by Khan Academy

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Key Questions

  • In order to conduct a one-sample proportion z-test, the following conditions should be met:

    1. The data are a simple random sample from the population of interest.
    2. The population is at least 10 times as large as the sample.
    3. #n*p>=10# and #n*(1-p)>=10#, where #n# is the sample size and #p# is the true population proportion.
  • A one sample z test for a proportion is a test meant to evaluate if a certain proportion #p# is equal or not to a certain value #p_0#.
    The test is calculated in this way:

    #z_0 = (hat p - p_0)/(sqrt((p_0*(1-p_0))/n)#

    where #hat p# is the proportion calculated on your sample.
    then, depending on which type of alternative hypotesis and value of #alpha# do you have, you accept or reject the null hypothesis depending on the value of #z_0#.

  • #\hatp# : the estimated value of the parameter #p# based on your sample.

    #\p_0# : the real value of #p# under the null hypothesis.

    p-value : formally, the probability, under the null hypothesis of obtaining the observed value or any other value wich brings more evidence to reject #H_0#.

    As an example let's say you want to test if a coin toss is actually fair or heads has a larger probability. In this case you would have:

    #H_0#: #p = p_0 = 0.5# vs #H_1#: #p > 0.5#, where p is the probability of the "heads" results.

    Then you calculate your #\hatp# wich in this case will be the proportion of heads in your sample. Let's say you got a 0.65 proportion in your sample.

    Then your p-value will be the probability under #H_0# (this means you calculate this probability assuming it's a fair coin) of having a proportion of 0.65 or more in your sample, because the larger the proportion, we get more evidence that "heads" has more probability.

    Small p-values leads us to reject #H_0#.

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