Question

does owning an indoor dog really help bring down borderline high systolic blood pressure? to answer this question, a random sample of twelve subjects with chronically borderline high systolic blood pressure was obtained. their systolic blood pressure was recorded. they then received a house-trained dog to live indoor with them. after one month, their blood pressure was taken again. at the 2.5% significance level, can you conclude that owning a pet will reduce the average systolic blood pressure? state the observed value of the test statistic.

Answer 1

Main answer:

At the 2.5% significance level, we conclude that owning a pet reduces the average systolic blood pressure.

The observed value of the test statistic is approximately 1.94.

Step-by-step explanation:

To determine if owning an indoor dog helps bring down borderline high systolic blood pressure, perform a paired t-test on the given data.

The null hypothesis (H₀) is that owning a pet does not reduce the average systolic blood pressure, and the alternative hypothesis (H₁) is that owning a pet does reduce the average systolic blood pressure.

Calculate the differences in blood pressure before and after owning a pet:

Differences: (139.9 - 136.3), (130.4 - 127.4), (131.3 - 125.6), (130.8 - 132.2), (128.9 - 127.5), (129.5 - 126.9), (130.8 - 131.1), (129.3 - 123.7), (128.1 - 130.4), (133.4 - 134.8), (125.2 - 125.6), (123.9 - 124.9)

Differences: 3.6, 3.0, 5.7, -1.4, 1.4, 2.6, -0.3, 5.6, -2.3, -1.4, -0.4, -1.0

Now, perform the paired t-test using these differences.

Since the sample size is small (n = 12) and the population standard deviation is unknown, we will use a t-distribution.

The observed value of the test statistic is calculated as follows:

t = (mean of differences) / (standard deviation of differences /
`\sqrt`(n))

Mean of differences = (3.6 + 3.0 + 5.7 - 1.4 + 1.4 + 2.6 - 0.3 + 5.6 - 2.3 - 1.4 - 0.4 - 1.0) / 12 ≈ 1.26

Standard deviation of differences =
`\sqrt`([sum of squares of differences - (sum of differences
`)^2` / n] / (n - 1))

Sum of squares of differences =
`(3.6^2 + 3.0^2 + 5.7^2 - 1.4^2 + 1.4^2 + 2.6^2 - 0.3^2 + 5.6^2 - 2.3^2 - 1.4^2 - 0.4^2 - 1.0^2) \approx 117.27`

12.96 + 9 + 32.49 + 1.96 + 1.96 + 6.76 + 0.09 + 31.36 + 5.29 + 1.96 + 0.16 + 1 = 95.99

Sum of differences = 3.6 + 3.0 + 5.7 - 1.4 + 1.4 + 2.6 - 0.3 + 5.6 - 2.3 - 1.4 - 0.4 - 1.0 ≈ 15.1

Standard deviation of differences =
`\sqrt`((95.997 - (15.1^2 / 12)) / (12 - 1)) ≈ ≈ 2.249

`\sqrt`(n) =
`\sqrt`(12) ≈ 3.464

t = 1.26 / (2.249 / 3.464) ≈ 1.94

To determine if the observed t-value is statistically significant at the 2.5% significance level, we need to compare it with the critical t-value for the given degrees of freedom (n - 1 = 12 - 1 = 11) and significance level.

Looking up the critical t-value in a t-distribution table or using statistical software, we find that the critical t-value for a one-tailed test with 11 degrees of freedom at a 2.5% significance level is approximately 1.796.

Since the observed t-value (1.94) is greater than the critical t-value (1.796), we reject the null hypothesis. Therefore, at the 2.5% significance level, we conclude that owning a pet reduces the average systolic blood pressure.

The observed value of the test statistic is approximately 1.94.

Does owning an indoor dog really help bring down borderline high systolic blood pressure? To answer this question, a random sample of twelve subjects with chronically borderline high sys- tolic blood pressure was obtained. Their systolic blood pressure was recorded. They then received a house-trained dog to live indoor with them. After one month, their blood pressure was taken again. At the 2.5% significance level, can you conclude that owning a pet will reduce the average systolic blood pressure? state the observed value of the test statistic.

Blood Pressure Before 139.9 130.4 131.3 130.8 128.9 129.5 130.8 129.3 128.1 133.4 125.2 123.9

Pressure After Blood 136.3 127.4 125.6 132.2 127.5 126.9 131.1 123.7 130.4 134.8 125.6 124.9