Popular wisdom is that eating pre-sweetened cereal tends to increase the
number of dental caries (cavities) in children. A sample of children was
(with parental consent) entered into a study and followed for several years.
Each child was classified as a sweetened-cereal lover or a non-sweetened
cereal lover. At the end of the study, the amount of tooth damage was
measured. Here is the summary data:
Group n mean std. dev
Sugar Bombed 10 6.41 5.0
No sugar 15 5.20 15.0
An approximate 95% confidence interval for the difference in the mean
tooth damage is:
ANS.
Degrees of Freedom
v=18.26
(6.41 − 5.20) ± 2.26(Sqrt.(25/10 + 225/15
i want to know where and how the 2.26 came from. I see this from the 2-tail .05 and df=9.
(6.41 − 5.20) ± 2.26(Sqrt.(25/10 + 225/15))
For 95% confidence interval, t with 9 df = ± 2.26.
Look on table for t probabilities.
The value 2.26 represents the critical value from the t-distribution for a 95% confidence level with degrees of freedom (df) equal to 9. This value is obtained from statistical tables or using statistical software.
In this case, the formula used to calculate the t-value is:
t = critical value * standard error
The critical value (2.26) is chosen based on the desired confidence level (95%) and the degrees of freedom (9). The standard error in this case is obtained by dividing the standard deviation by the square root of the sample size.
So, the formula used to calculate the confidence interval is:
(6.41 - 5.20) ± 2.26 * (sqrt((25/10) + (225/15)))
The numerator of each term in the square root is the sample size of each group divided by their respective degrees of freedom.
By substituting the values, we obtain the confidence interval for the difference in the mean tooth damage.
To understand where the value of 2.26 came from, let's break down the steps involved in calculating the confidence interval for the difference in the mean tooth damage.
Step 1: Calculate the Standard Error
The first step is to calculate the standard error, which measures the variability in the sample means. In this case, we have two different groups (sugar bombed and no sugar).
The formula for calculating the standard error is:
SE = sqrt[(s1^2/n1) + (s2^2/n2)]
where s1 and s2 are the standard deviations of the two groups, and n1 and n2 are the sample sizes.
From the given data, we have:
s1 = 5.0 (standard deviation of the sugar bombed group)
s2 = 15.0 (standard deviation of the no sugar group)
n1 = 10 (sample size of the sugar bombed group)
n2 = 15 (sample size of the no sugar group)
Plugging in the values:
SE = sqrt[(5.0^2/10) + (15.0^2/15)]
= sqrt[(25/10) + (225/15)]
= sqrt[2.5 + 15]
= sqrt(17.5)
≈ 4.1833
Step 2: Calculate the t-value
The t-value is obtained from the t-distribution table based on the desired significance level and the degrees of freedom (df). In this case, you mentioned using a 2-tail significance level of 0.05 and df = 9.
To find the t-value, you would look up the critical value in the t-distribution table for a 2-tail test with 9 degrees of freedom and a significance level of 0.05 (split between two tails). Looking up these values, you will find that the t-value is approximately 2.26.
Step 3: Calculate the Margin of Error
The margin of error represents the range within which the true population mean difference is likely to lie.
The formula for calculating the margin of error is:
Margin of Error = t-value * SE
Plugging in the values:
Margin of Error = 2.26 * 4.1833
≈ 9.45
Step 4: Calculate the Confidence Interval
Finally, you can calculate the confidence interval by subtracting and adding the margin of error to the difference in means.
Given that the mean tooth damage for the sugar bombed group is 6.41 and for the no sugar group is 5.20, the estimated difference in means is (6.41 - 5.20) = 1.21.
Confidence Interval = (1.21 - 9.45, 1.21 + 9.45)
= (-8.24, 10.67)
Thus, the approximate 95% confidence interval for the difference in the mean tooth damage is (-8.24, 10.67).
In summary, the value of 2.26 is the critical t-value obtained from the t-distribution table for a 2-tail 0.05 significance level and 9 degrees of freedom. It is used to calculate the margin of error, which is then used to construct a confidence interval for the difference in the mean tooth damage between the two groups.