Question

1. When we conduct hypothesis testing with proportions, we have a sample size, n, and a number of successes, x. We compare our sample information to a previously collected population proportion and perform a hypothesis test to determine whether to retain or reject the null hypothesis. Suppose we collect n = 100 data points and find that of the 100 people sampled, x = 15 are left handed. We are then asked to perform a hypothesis test to determine whether there is sufficient evidence to suggest that the proportion of left-handed people has increased from the previous measured value of 11% of the total population. WHY can't we simply say that 15/100 = 0.15 = 15% which is higher than 11% and end it right there? Why is conducting a hypothesis test necessary to answer this question? What does a hypothesis test do that a simple comparison of the two numbers doesn't do?

2. We use very specific vocabulary when we conduct hypothesis tests. We speak of "failing to reject the null" or "retaining the null". Why is this so? Do hypothesis tests PROVE the null or do they simply FAIL to disprove the null? Explain.

Answer 1

1)

Main reason is because we’re looking at a sample rather than the entire population.

Sampling error is the difference between a sample and the entire population. Due to sampling error, it’s entirely possible that our sample proportion may different in each sample. if we repeated the experiment, it’s possible that the second sample proportion might be different or greater tha 15%. A hypothesis test helps assess the likelihood of this possibility and considered the standard errror to counter the same.

2)

Null hypothesis are never accepted. We either reject them or fail to reject them. The distinction between “acceptance” and “failure to reject” is best understood in terms of confidence intervals. Failing to reject a hypothesis means a confidence interval contains a value of “no difference”.

Please revert in case of any doubt.

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