When the tissue around teeth becomes inflamed, the tissue bleeds more easily and this is an indication of
gum disease. The severity of bleeding can be measured by a bleeding score that is determined by the
percentage of teeth for which bleeding occurs when the tissue surrounding the tooth is probed. In terms of
dental health, lower bleeding scores are better than higher bleeding scores. The authors of a paper
wondered if the nutrients found in kiwis might reduce bleeding. To investigate, 25 adult participants agreed
to consume 2 kiwis a day for a period of 2 months. A bleeding score was recorded for each participant at
the beginning of the study period and again after 2 months of kiwi consumption. Because this resulted in
paired samples, differences in bleeding scores (before − after) were analyzed. It was reported that the
sample mean difference was xd = 6.64 and the standard deviation of the differences was sd = 11.8
Assume that the 25 study participants can be considered as representative of the population of adults
Use the information given above to construct a 90% confidence interval for the difference in bleeding score
before and after 2 months of kiwi consumption. (Use Before − After. Enter your answer using interval
notation. Round your numerical values to four decimal places.)
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To construct a 90% confidence interval for the difference in bleeding score before and after 2 months of kiwi consumption, we can use the formula:
CI = xd ± t * (sd / sqrt(n))
Where:
CI = Confidence Interval
xd = Sample mean difference
t = t-value for a 90% confidence level with (n - 1) degrees of freedom (df)
sd = Standard deviation of the differences
n = Sample size
Given:
xd = 6.64
sd = 11.8
n = 25
We need to find the t-value for a 90% confidence level with (n - 1) degrees of freedom. Since n = 25, df = 25 - 1 = 24. Using a t-table or calculator, the t-value for a 90% confidence level with 24 degrees of freedom is approximately 1.7109.
Substituting the values into the formula:
CI = 6.64 ± 1.7109 * (11.8 / sqrt(25))
Calculating the value within the parentheses:
(11.8 / sqrt(25)) ≈ 2.36
Calculating the confidence interval:
CI = 6.64 ± 1.7109 * 2.36
CI = 6.64 ± 4.0364
CI ≈ (2.6036, 10.6764)
Therefore, the 90% confidence interval for the difference in bleeding score before and after 2 months of kiwi consumption is (2.6036, 10.6764).