Question

homework
question 6, 9.2.11 - t
part 6 of 7
hw score: 55.86%, 6.7 of 12 points
points: 0.53 of 1
researchers conducted a study to determine whether magnets are effective in treating back pain. pain was measured using the visual analog scale, and the results shown below are among the results obtained in the study. higher scores correspond to greater pain levels. assume that the two samples are independent simple random samples selected from normally distributed populations, and do not assume that the population standard deviations are equal. complete parts (a) to (c) below.
reduction in pain level after magnet treatment (μ₁) n = 15, x̄ = 0.57, s = 0.89
reduction in pain level after sham treatment (μ₂) n = 15, x̄ = 0.53, s = 1.56
a. h₀: μ₁ = μ₂
h₁: μ₁ > μ₂
b. h₀: μ₁ ≠ μ₂
h₁: μ₁ < μ₂
c. h₀: μ₁ = μ₂
h₁: μ₁ ≠ μ₂
d. h₀: μ₁ < μ₂
h₁: μ₁ ≥ μ₂
the test statistic, t, is 0.09. (round to two decimal places as needed.)
the p - value is 0.465. (round to three decimal places as needed.)
state the conclusion for the test.
fail to reject the null hypothesis. there is not sufficient evidence to support the claim that those treated with magnets have a greater mean reduction in pain than those given a sham treatment.
b. construct a confidence interval appropriate for the hypothesis test in part (a).

• 0.76 < μ₁ - μ₂ < 0.84 (round to two decimal places as needed.)

c. does it appear that magnets are effective in treating back pain? is it valid to argue that magnets might appear to be effective if the sample sizes are larger? choose the correct answer below.
a. it appears that magnets are not effective in treating back pain, because 0 is in the confidence interval.
b. it appears that magnets are not effective in treating back pain, because the p - value is less than the significance level.
c. it appears that magnets are effective in treating back pain, because the p - value is greater than the significance level.
d. it appears that magnets are effective in treating back pain, because the confidence interval contains only positive values.
get more help -
an example
clear all
check answer
5:41 pm
12/8/2025

Explanation:

Step1: Identify the Hypotheses

We want to test if magnets are effective, meaning the mean pain reduction for magnet treatment ($\mu_1$) is greater than for sham treatment ($\mu_2$). So the null hypothesis $H_0: \mu_1 = \mu_2$ and alternative hypothesis $H_1: \mu_1 > \mu_2$. So option A is correct for part (a).

Step2: Interpret P - value for Conclusion

The P - value is 0.465. Typically, we use a significance level like $\alpha = 0.05$. Since $0.465>0.05$, we fail to reject the null hypothesis. So there is not sufficient evidence to support the claim that magnet - treated patients have a greater mean pain reduction.

Step3: Interpret Confidence Interval

The confidence interval for $\mu_1-\mu_2$ is $- 0.76<\mu_1 - \mu_2<0.84$. This interval contains 0, which means we can't be sure that $\mu_1>\mu_2$ (if the interval was entirely positive, we could be more confident). But the P - value (0.465) is greater than a typical significance level (e.g., 0.05), so we fail to reject $H_0$. For the second part of (c), if sample sizes were larger, the standard error would decrease (because standard error $=\frac{s}{\sqrt{n}}$), which could lead to a smaller P - value (more significant result) or a confidence interval that doesn't contain 0, so it might be valid to argue magnets are effective with larger samples. But based on the given sample sizes, the answer for the first part of (c) is that it does not appear magnets are effective (option A or B? Wait, the P - value is 0.465, which is large, so we fail to reject $H_0$. So "It appears that magnets are not effective in treating back pain, because the P - value is greater than the significance level" (but wait, the P - value is 0.465, which is greater than 0.05, so we fail to reject. So the reasoning: if P - value > $\alpha$, fail to reject $H_0$. So the correct option for (c) first part: Let's check the options. Option A: "It appears that magnets are not effective in treating back pain, because 0 is in the confidence interval" (the confidence interval contains 0, which means $\mu_1-\mu_2$ could be 0, so $\mu_1=\mu_2$). Option B: "It appears that magnets are not effective in treating back pain, because the P - value is less than the significance level" (no, 0.465 is not less than 0.05). Option C: "It appears that magnets are effective... P - value greater than..." (no, if P - value > $\alpha$, we don't have evidence for $H_1$). Option D: "confidence interval contains only positive values" (no, the interval has negative and positive values). So option A is correct for the first part of (c) because the confidence interval contains 0, meaning there's no significant difference.

Step4: Larger Sample Sizes

If sample sizes are larger, the power of the test increases (ability to detect a true difference). A larger sample size reduces the standard error, so the test statistic (t - statistic) would be more likely to be in the rejection region if there is a true difference. So it might be valid to argue magnets are effective with larger samples.

Answer:

Part (a)

A. $H_0:\mu_1 = \mu_2$, $H_1:\mu_1>\mu_2$

Part (b)

Fail to reject the null hypothesis. There is not sufficient evidence to support the claim that those treated with magnets have a greater mean reduction in pain than those given a sham treatment.

Part (c)

• First part: A. It appears that magnets are not effective in treating back pain, because 0 is in the confidence interval.
• Second part: Yes, it might be valid to argue that magnets might appear to be effective if the sample sizes are larger because larger sample sizes increase the power of the test, making it more likely to detect a true difference (if it exists) between the two population means.