A researcher wants to determine if two brands of cigarettes contain the same amount of nicotine. The data is shown in the table. With a significance level of 0.05 to that conclusion can be reached. Set the interval for a confidence of 95%
Sample A: 40.3 37.5 37 41.7 29.8 33.5
31.3 33.9 34.7 30.3 39.8 40.9 32.7 39.2
38.4 25.1 32.2 38 32.8 31.8
Sample B: 46.9 47.6 41.6 41.3 36.5 39.8
45.3 43.1 42.1 36.8 46.4 41.2 47.3 45.5
34.9 41.0 37.3 42.7 38.0 35.7 34.6 38.0
41.0 41.6 Assume That the two Populations Are Normally Distributed.
To determine if the two brands of cigarettes contain the same amount of nicotine, we can use a two-sample t-test. In this case, we will set a significance level of 0.05.
First, let's calculate the mean and standard deviation for each sample:
Sample A:
Mean (x̄A) = (40.3 + 37.5 + 37 + 41.7 + 29.8 + 33.5 + 31.3 + 33.9 + 34.7 + 30.3 + 39.8 + 40.9 + 32.7 + 39.2 + 38.4 + 25.1 + 32.2 + 38 + 32.8 + 31.8) / 20 = 35.805
Standard Deviation (sA) = sqrt([(40.3 - 35.805)^2 + (37.5 - 35.805)^2 + ... + (31.8 - 35.805)^2] / (20 - 1))
Sample B:
Mean (x̄B) = (46.9 + 47.6 + 41.6 + 41.3 + 36.5 + 39.8 + 45.3 + 43.1 + 42.1 + 36.8 + 46.4 + 41.2 + 47.3 + 45.5 + 34.9 + 41.0 + 37.3 + 42.7 + 38.0 + 35.7 + 34.6 + 38.0 + 41.0 + 41.6) / 23 = 41.488
Standard Deviation (sB) = sqrt([(46.9 - 41.488)^2 + (47.6 - 41.488)^2 + ... + (41.6 - 41.488)^2] / (23 - 1))
Next, we need to calculate the t-statistic using the formula:
t = (x̄A - x̄B) / sqrt(sA^2/nA + sB^2/nB)
where nA is the sample size of Sample A and nB is the sample size of Sample B.
Finally, with the t-statistic, we can determine the p-value to determine if the conclusion can be reached. The p-value represents the probability of observing the data if the null hypothesis is true (the null hypothesis in this case being that the two brands of cigarettes contain the same amount of nicotine).
Using the p-value, if it is less than the significance level (0.05 in this case), we can reject the null hypothesis and conclude that there is a significant difference in nicotine content between the two cigarette brands.
As for setting the interval for a confidence of 95%, it seems like you may be looking for the confidence interval for the difference in means. To do this, we can use the formula:
(x̄A - x̄B) ± t(Two-Tailed) * sqrt(sA^2/nA + sB^2/nB)
where t(Two-Tailed) is the critical value for a desired confidence level. For a confidence level of 95%, we would use a critical value of 2.09 (assuming a large sample size).
This confidence interval will provide an estimate of the true difference in means and its range within which we can be confident the true difference lies.