Q. 9.117 - Elementary Statistics

Question

Ankle Brachial Index. The ankle brachial index (ABI) compares the blood pressure of a patient's arm to the blood pressure of the patient's leg. The ABI can be an indicator of different diseases, including arterial diseases. A healthy (or normal) ABI is 0.9 or greater. In a study by M. McDermott et al. titled "Sex Differences in Peripheral Arterial Disease: Leg Symptoms and Physical Functioning" (Journal of the American Geriatrics Society, Vol. 51, No. 2, Pp. 222-228), the researchers obtained the ABI of 187 women with peripheral arterial disease. The results were a mean ABI of 0.64 with a standard deviation of 0.15 At the 1 % significance level, do the data provide sufficient evidence to conclude that, on average, women with peripheral arterial disease have an unhealthy ABI?

Step-by-Step Solution

Answer

From the MINITAB output, the P-value is 0.000

Step 1: Given Information

Check whether the data provide sufficient evidence to conclude that on average, women with peripheral arterial disease have an unhealthy ABI.

State the null and alternative hypothesis:

Null hypothesis:

$H_o:\mu =0.9$

That is, the data does not provide sufficient evidence to conclude that on average, women with peripheral arterial disease have an unhealthy ABI. 

Alternative hypothesis:

$H_o:\mu <0.9$

That is, the data provide sufficient evidence to conclude that on average, women with peripheral arterial disease have an unhealthy ABI

Decide a significance level

Here, the significance level is, $\alpha =0.01$

Step 2: Explanation

Compute the value of the test statistic and P-value by using MINITAB.

MINITAB procedure:

Step 1: Choose Stat > Basic Statistics > 1-Sample t.

Step 2: In Summarized data, enter the sample size 187 and mean 0.64.

Step 3: In Standard deviation, enter a value 0.15.

Step 4: In Perform hypothesis test, enter the test mean as 0.9

Step 5: Check Options, enter Confidence level as 99

Step 6: Choose less than in alternative.

Step 7: Click OK in all dialogue boxes.

MINITAB output:

One-Sample T

Test of mu$=0.9 vs < 0.9$

99% Upper
N	Mean	StDev	SE Mean	Bound	T	P
187	0.6400	0.1500	0.0110	0.6657	-23.70	0.000

From the MINITAB output, 

The value of test statistic is $-23.70$

The P-value is $0.000$

Step 3: Conclusion

If $P\le \alpha$, then reject the null hypothesis.

Here, the P-value is 0.000 which is less than the level of significance. That is,

$P(=0.000)<\alpha (=0.01).$

Therefore, the null hypothesis is rejected at $1\%$ level.

Thus, it can be conclude that the test results are statistically significant at $1\%$ level of significance.

Interpretation:

Thus, the data provide sufficient evidence to conclude that on average, women with peripheral arterial disease have an unhealthy ABI