7.E: Introduction to Hypothesis Testing (Exercises)
- Page ID
- 7124
- In your own words, explain what the null hypothesis is.
- Answer:
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Your answer should include mention of the baseline assumption of no difference between the sample and the population.
- What are Type I and Type II Errors?
- What is \(α\)?
- Answer:
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Alpha is the significance level. It is the criteria we use when decided to reject or fail to reject the null hypothesis, corresponding to a given proportion of the area under the normal distribution and a probability of finding extreme scores assuming the null hypothesis is true.
- Why do we phrase null and alternative hypotheses with population parameters and not sample means?
- If our null hypothesis is “\(H_0: μ = 40\)”, what are the three possible alternative hypotheses?
- Answer:
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\(H_A: μ ≠ 40\), \(H_A: μ > 40\), \(H_A: μ < 40\)
- Why do we state our hypotheses and decision criteria before we collect our data?
- When and why do you calculate an effect size?
- Answer:
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We calculate an effect size when we find a statistically significant result to see if our result is practically meaningful or important
- Determine whether you would reject or fail to reject the null hypothesis in the following situations:
- \(z\)= 1.99, two-tailed test at \(α\) = 0.05
- \(z\) = 0.34, \(z*\) = 1.645
- \(p\) = 0.03, \(α\) = 0.05
- \(p\) = 0.015, \(α\) = 0.01
- You are part of a trivia team and have tracked your team’s performance since you started playing, so you know that your scores are normally distributed with \(μ\) = 78 and \(σ\) = 12. Recently, a new person joined the team, and you think the scores have gotten better. Use hypothesis testing to see if the average score has improved based on the following 8 weeks’ worth of score data: 82, 74, 62, 68, 79, 94, 90, 81, 80.
- Answer:
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Step 1: \(H_0: μ = 78\) “The average score is not different after the new person joined”, \(H_A: μ > 78\) “The average score has gone up since the new person joined.”
Step 2: One-tailed test to the right, assuming \(α\) = 0.05, \(z*\) = 1.645.
Step 3: \(\overline{\mathrm{X}}\)= 88.75, \(\sigma _{\overline{\mathrm{X}}}\) = 4.24, \(z\) = 2.54.
Step 4: \(z > z*\), Reject \(H_0\). Based on 8 weeks of games, we can conclude that our average score (\(\overline{\mathrm{X}}\) = 88.75) is higher now that the new person is on the team, \(z\) = 2.54, \(p\) < .05. Since the result is significant, we need an effect size: Cohen’s \(d\) = 0.90, which is a large effect.
- You get hired as a server at a local restaurant, and the manager tells you that servers’ tips are $42 on average but vary about $12 (\(μ\) = 42, \(σ\) = 12). You decide to track your tips to see if you make a different amount, but because this is your first job as a server, you don’t know if you will make more or less in tips. After working 16 shifts, you find that your average nightly amount is $44.50 from tips. Test for a difference between this value and the population mean at the \(α\) = 0.05 level of significance.