Week 11 Chapter 17. Testing Hypotheses about Proportions

Hypothesis Testing Access and Support to Education and Training Survey, in 2008, asked a random sample of 4,756 adult Canadians “Did (you/your spouse or partner) work at a job or business at any time (between July 2007 and June 2008)? (regardless of the number of hours per week)”. Of the 4,756 respondents, 1,581 answered yes. The above statement serves as a hypothesis and moreover as Research Hypothesis. A hypothesis is: a statement about a population. a predication that a parameter describing some characteristics of a variable (e.g., true proportion, 𝑝) takes a particular numerical value or falls in a certain range of values. For conducting a Hypothesis Test: Researchers (you) use data to summarize the evidence about a hypothesis. With data, you can compare the point estimates of parameters to the values predicted by the hypothesis.

Example: Hypothesis Testing for a Proportion, 𝒑 Suppose it was claimed that the percentage of adult Canadians who worked at a job or business at anytime (between July 2007 and June 2008), regardless of the number of hours per week, was 50%. Researchers claimed that the percentage of adult Canadians who worked at a job or business at anytime (between July 2007 and June 2008), regardless of the number of hours per week, is different from 50%. In order to test their hypothesis, these researchers relied on the obtained statistics from the Access and Support to Education and Training Survey, 2008, for a random sample of 4,756 adult Canadians. Of the 4,756 respondents, 1,581 indicated that they worked at a job or business at anytime (between July 2007 and June 2008), regardless of the number of hours per week . 𝑝 = 1581 4756 = 0.332 𝑝 is approx. Normal (mean: 𝑝 = 0.50, standard error: 𝜎 𝑝 = 𝑝(1−𝑝) 𝑛 = 0.50(1−0.50) 4756 = 0.007)

Hypothesis Testing for a Proportion (Two-sided Test) – Step 1 Step 1. Set up the null and alternative hypothesis: The null hypothesis is the current belief: 𝐻 0 : 𝑝= 𝑝 𝑂 In our example it would have a form: Ho: 𝑝 = 0.50 The Alternative hypothesis is what the researcher(s) [you] want to prove: 𝐻 𝑎 : 𝑝≠ 𝑝 𝑂 In our example it would have a form: 𝐻 𝑎 : 𝑝≠0.50. This means a two-sided test: either 𝑝<0.50 or 𝑝>0.50 * The goal here is to provide evidence against Ho (e.g., suggest Ha).

Hypothesis Testing for a Proportion (Two-sided Test) - Step 2 Step 2. Calculate the Test Statistic: Recall: 𝑝 = 1581 4756 = 0.332, 𝑝 is approx. Normal (mean: 𝑝 = 0.50, standard error: 𝜎 𝑝 = 𝑝(1−𝑝) 𝑛 = 0.50(1−0.50) 4756 = 0.007) Under Ho, the test statistics has a Z, standard normal distribution: Z = 𝑝 −𝑝 𝜎 𝑝 = 0.332 −0.500 0.50(1−0.50) 4756 = 0.332 −0.500 0.0073 = - 23.02

Hypothesis Testing for a Proportion (Two-sided Test) - Step 3 Step 3. Find P-value: P-value: The chance (the proportion) of getting a 𝑝 as far as or further from 𝐻 0 than the value observed; That means, P-value is the probability of getting at least something (e.g., sample proportion, 𝑝 ) more extreme (e.g., unusual, unlikely, or rare) than what we have already found (our observed value of 𝑝 ) that provide even stronger evidence against Ho. The more extreme the z-score (large in absolute values) are the ones that denote farther departure of the observed value (e.g., our 𝑝 ) from the parameter value ( 𝑝 𝑜 ) in Ho. In the two-sided test, e.g., 𝑯 𝒂 : 𝒑≠ 𝒑 𝑶 , P-Value is the two-tailed probability. This is the probability that sample proportion 𝑝 falls at least as far from 𝑝 𝑜 in either direction as the observed value of 𝑝 . In our example: P-value = 2 x (Probability less than Z = -23.02) = 2 (≈ 0) ≅ 0 (or better to say less than 0.0001)

Hypothesis Testing for a Proportion (Two-sided Test) - Step 4 Step 4. Conclusion: We use the P-Value for stating our conclusion to our research question. When Ho is true, P-Value is roughly equally likely to fall anywhere between 0 and 1. When Ho is false, P-Value is closer to 0 than 1. The smaller the P-Value the stronger evidence we have against Ho and thus we have stronger evidence to support Ha (e.g., sufficient evidence to conclude our claim). But how small is small? We would need to choose an 𝛼-level: a number such that if: P-value ≤ 𝛼-level, we reject Ho; We can conclude Ha (we have evidence to support our claim). Often we phrase as a statistically significant result at that specified 𝛼-level. P-value >𝛼-level, we fail to reject Ho; We cannot conclude Ha (we have not enough evidence to support our claim; thus, Ho is plausible – We do not accept Ho). Often we phrase as the result is not statistically significant at that specified 𝛼-level. The default 𝛼-level, 𝛼 = 0.05 So, in our example, Conclusion is: P-value (approx. 0) < 𝜶 = 0.05; We reject Ho and conclude Ha that the result is statistically significant at 𝛼 = 0.05. We have strong evidence to conclude that the true percentage of adult Canadians who worked at a job or business at anytime (between July 2007 and June 2008), regardless of the number of hours per week, was different from 50%.

Hypothesis Testing for a Proportion (Two-sided Test) – All Steps Ho: 𝑝 = 0.50 Ha: 𝑝 ≠ 0.50 Under Ho, the test statistics has a Z, standard normal distribution: Z = 𝑝 −𝑝 𝜎 𝑝 = 0.332 −0.500 0.50(1−0.50) 4756 = 0.332 −0.500 0.0073 = - 23.02 Recall: 𝑝 = 1581 4756 = 0.332 P-value = 2 x (Probability less than Z = -23.02) = 2 (≈ 0) ≅ 0 (or better to say less than 0.0001) Conclusion: P-value < 𝛼 = 0.05; Reject Ho and conclude Ha. We have strong evidence to conclude that the true percentage of adult Canadians who worked at a job or business at anytime (between July 2007 and June 2008), regardless of the number of hours per week, was different from 50%.

Hypothesis Testing for a Proportion (Two-sided Test) – StatCrunch Command Stat> Proportion Stats> One Sample > With Summary # of success: 1581 # of observation: 4756 Perform Hypothesis test for P H0 : p = 0.5 HA : p ≠ 0.5 Click on Compute

Hypothesis Testing for a Proportion (Two-sided Test) – StatCrunch Output One sample proportion summary hypothesis test: p : Proportion of successes H0 : p = 0.5 HA : p ≠ 0.5 Conclusion: Z = -23.11, P-value < 0.0001 (which is less than 𝛼 = 0.05); We reject Ho and conclude Ha; We have strong evidence to conclude that the true percentage of adult Canadians who worked at a job or business at anytime (between July 2007 and June 2008), regardless of the number of hours per week, was different from 50%. Proportion Count Total Sample Prop. Std. Err. Z-Stat P-value p 1581 4756 0.3324222 0.0072501849 -23.113589 <0.0001