Nicotine (mg) Filtered Kings
Nonfiltered Kings
n1 = 21
n2 = 8
sample mean = 0.94
sample mean = 1.65
s = 0.31
s = 0.16
Suppose you were to conduct a test (at the 0.05 significance level) to test the claim that king-size cigarettes with filters have a lower mean amount of nicotine than the mean amount of nicotine in non-filtered king-size cigarettes.
For this test, the critical value (to 3 decimal places) is
and the test statisic (to 2 decimal places) is .
You would use a one-tailed test for the difference between means. The difference between the means needs to be divided by the standard error (SE) of the difference between means.
SE of the difference between means equals the square root of the sum of the squared SEs of the two means.
The SE for each mean is the standard deviation (s) divided by the square root of n-1.
I will leave the calculations to you.
Look up the .05 value in the smaller area in a table of areas under the normal distribution in the back of your text. This will help you find the critical value you seek.
I hope this helps. Thanks for asking.
To conduct a test to compare the means of two populations, in this case, the mean amount of nicotine in filtered king-size cigarettes and non-filtered king-size cigarettes, we can use a two-sample t-test.
First, let's define our null and alternative hypotheses:
Null Hypothesis (H0): The mean amount of nicotine in filtered king-size cigarettes is equal to or greater than the mean amount of nicotine in non-filtered king-size cigarettes. (μ1 >= μ2)
Alternative Hypothesis (Ha): The mean amount of nicotine in filtered king-size cigarettes is lower than the mean amount of nicotine in non-filtered king-size cigarettes. (μ1 < μ2)
Next, we need to calculate the test statistic. The formula for the two-sample t-test is:
t = (x̄1 - x̄2) / √((s1^2 / n1) + (s2^2 / n2))
where x̄1 and x̄2 are the sample means, s1 and s2 are the sample standard deviations, and n1 and n2 are the sample sizes of the two groups.
In this case:
x̄1 = 0.94
x̄2 = 1.65
s1 = 0.31
s2 = 0.16
n1 = 21
n2 = 8
Plugging these values into the formula, we get:
t = (0.94 - 1.65) / √((0.31^2 / 21) + (0.16^2 / 8))
Next, we calculate the degrees of freedom (df) using the formula:
df = (s1^2 / n1 + s2^2 / n2)^2 / ((s1^2 / n1)^2 / (n1 - 1) + (s2^2 / n2)^2 / (n2 - 1))
Substituting the values, we get:
df = (0.31^2 / 21 + 0.16^2 / 8)^2 / ((0.31^2 / 21)^2 / (21 - 1) + (0.16^2 / 8)^2 / (8 - 1))
Finally, we need to find the critical value for our desired significance level (α) of 0.05. The critical value corresponds to the t-distribution with the appropriate degrees of freedom.
Using a t-table or a statistical calculator, we find the critical value for a one-tailed test at α = 0.05 and the degrees of freedom we calculated.
The critical value is the value that separates the rejection region (the area where we reject the null hypothesis) from the non-rejection region (the area where we fail to reject the null hypothesis).
Once you have the critical value and the test statistic, you can compare them. If the test statistic falls in the rejection region (less than the critical value), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Unfortunately, without the actual values of the critical value and the degrees of freedom, I cannot provide those specific answers. You can calculate the critical value and the degrees of freedom using the formulas provided above and consult a t-table or use a statistical calculator to find the precise values.