Question

(1) A researcher wants to test H0: x1 = x2 versus the two-sided alternative Ha: x1 ≠ x2.

(a) This alternative hypothesis indicates a one-sided hypothesis instead of a two-sided hypothesis.

(b) The alternative hypothesis Ha should indicate that x1 ≥ x2 .

(c)The null hypothesis (but not the alternative hypothesis) should involve μ₁ and μ₂ (population means) rather than x₁ and x₂ (sample means).

(d) Hypotheses should involve μ₁ and μ₂ (population means) rather than x₁ and x₂ (sample means).

(e) The null hypothesis H₀ should indicate that the two means are not equal.

(2) A study recorded the IQ scores of 50 college freshmen. The scores of the 24 males in the study were compared with the scores of all 50 freshmen using the two-sample methods of this section.

(a)The samples are too large to be used for hypothesis testing.

(b)The samples are not independent; we would need to compare the 24 males to the 26 females.    

(c) The samples are too small to be used for hypothesis testing.

(d)The sample sizes are too different to be used for hypothesis testing; we would need to have more males in the sample.

(e)A two-sample method is not appropriate in this situation.

(3)A two-sample t statistic gave a P-value of 0.93. From this we can reject the null hypothesis with 90% confidence.

(a)We can reject the null hypothesis, but with less than 90% confidence.

(b) We need the P-value to be small to reject H₀.    

(c) We can reject the null hypothesis, but with more than 90% confidence.

(d) We need the P-value to be negative to reject H₀.

(e)A P-value of this size is impossible.

(4)A researcher is interested in testing the one-sided alternative H_a: μ₁ < μ₂. The significance test gave t = 2.25. Since the P-value for the two-sided alternative is 0.04, he concluded that his P-value was 0.02.

(a) The alternative hypothesis should state that H_a: μ₁ ≠ μ₂.

(b) A one-sided alternative should never be used.    

(c) The alternative hypothesis should state that H_a: μ₁ ≤ μ₂.

(d) Assuming the researcher computed the t statistic using x₁ − x₂,a positive value of t does not support H_a.

(e)A t statistic of this size should have a much larger P-value associated with it.

Answer 1

(1) and are sample means. When testing a Hypothesis, we are testing if there are any changes in the population means ( and ), by taking samples and testing them.

Therefore: Option d: Hypotheses should involve and (population means) rather than and (sample means).

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(2) When we compare a smaller sample fom a larger population, we do not use a 2 sample t test as the samples will not be independent, and hence we cannot do an independent t test of 2 samples.

Therefore Option e: A two sample method is not appropriate in this situation.

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(3) To Reject the Null Hypothesis, the decision rule is; If p value is < , Then Reject H0. In Normal Hypothesis Testing cases, we use the following values of = 0.10, 0.05 or 0.01. Under extreme cases, we also use 0.001. To reject H0, you p value has to be lesser than these, and to reject H0 with 90% confidence, does not mean that you are 90% confident of rejecting H0, when p is large.

Therefore Option b: We need the P - value to be small to Reject H0.

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(4) If the researcher is interested in testing Ha: < , then the difference of the sample means given by - , should be negative. If - , the test will never give a significant result as the Observed value w(positive) will always be greater than the left tailed critical value (negative), and we would always fail to reject H0.

Therefore Option d: Assuming the researcher computed the t statistic using - , a positive value of t does not support Ha.

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