Question

migraine and acupuncture: a migraine is a particularly painful type of headache, which patients sometimes wish to treat with acupuncture. to determine whether acupuncture relieves migraine pain, researchers conducted a randomized controlled study where 158 patients diagnosed with migraine headaches were randomly assigned to one of two groups: treatment or control. 68 patients in the treatment group received acupuncture that is specifically designed to treat migraines. 90 patients in the control group received placebo acupuncture (needle insertion at non - acupoint locations). 24 hours after patients received acupuncture, they were asked if they were pain free. results are summarized in the contingency table below. test to see if migraine pain relief is dependent on receiving acupuncture. use \\( \alpha = 0.01 \\).

a) what is the correct null hypothesis?
\\( \bigcirc h _ { 0 } : p _ { 1 } = p _ { 2 } = p _ { 3 } = p _ { 4 } \\)
\\( \bigcirc h _ { 0 } : \\) migraine pain relief is dependent on receiving acupuncture.
\\( \bigcirc h _ { 0 } : \\) migraine pain relief is independent on receiving acupuncture.
\\( \bigcirc h _ { 0 } : p _ { 1 } \
eq p _ { 2 } \
eq p _ { 3 } \
eq p _ { 4 } \\)

b) fill in the expected values, round answers to at least 4 decimal places.

c) find the test statistic, round answer to at least 4 decimal places.

d) what is the p - value?

e) what is the correct conclusion?
\\( \bigcirc \\) there is statistical evidence that migraine pain relief is dependent on receiving acupuncture.
\\( \bigcirc \\) there is no statistical evidence that migraine pain relief is dependent on receiving acupuncture.

Explanation:

Step1: Calculate the expected values

The formula for the expected value \(E_{ij}=\frac{(R_i\times C_j)}{N}\), where \(R_i\) is the row total, \(C_j\) is the column total, and \(N\) is the grand total.
For the treatment - pain free yes cell: \(E_{11}=\frac{68\times29}{158}\approx12.5316\)
For the treatment - pain free no cell: \(E_{12}=\frac{68\times129}{158}\approx55.4684\)
For the control - pain free yes cell: \(E_{21}=\frac{90\times29}{158}\approx16.4684\)
For the control - pain free no cell: \(E_{22}=\frac{90\times129}{158}\approx73.5316\)

Step2: Calculate the test statistic

The test statistic for a chi - square test of independence is \(\chi^{2}=\sum\frac{(O - E)^{2}}{E}\), where \(O\) is the observed value and \(E\) is the expected value.
\(\chi^{2}=\frac{(20 - 12.5316)^{2}}{12.5316}+\frac{(48 - 55.4684)^{2}}{55.4684}+\frac{(9 - 16.4684)^{2}}{16.4684}+\frac{(81 - 73.5316)^{2}}{73.5316}\)
\(=\frac{7.4684^{2}}{12.5316}+\frac{- 7.4684^{2}}{55.4684}+\frac{-7.4684^{2}}{16.4684}+\frac{7.4684^{2}}{73.5316}\)
\(=\frac{55.777}{12.5316}+\frac{55.777}{55.4684}+\frac{55.777}{16.4684}+\frac{55.777}{73.5316}\)
\(\approx4.451+1.006+3.388 + 0.759\approx9.604\)

Step3: Determine the p - value

The degrees of freedom \(df=(r - 1)(c - 1)\), where \(r\) is the number of rows and \(c\) is the number of columns. Here \(r = 2\) and \(c=2\), so \(df=(2 - 1)(2 - 1)=1\)
Using a chi - square distribution table or a calculator with a chi - square cdf function (\(\text{X}^{2}\text{cdf}(9.604,\infty,1)\)), the p - value is approximately \(0.0019\)

Step4: Make a conclusion

Since the p - value (\(0.0019\)) is less than \(\alpha = 0.01\), we reject the null hypothesis.

Answer:

a) \(H_{0}\): Migraine pain relief is independent on receiving acupuncture.
b) Treatment - Pain Free: Yes: \(12.5316\), Treatment - Pain Free: No: \(55.4684\), Control - Pain Free: Yes: \(16.4684\), Control - Pain Free: No: \(73.5316\)
c) \(\chi^{2}\approx9.604\)
d) \(p - value\approx0.0019\)
e) There is statistical evidence that migraine pain relief is dependent on receiving acupuncture.