What impact does fast-food consumption have on various dietary and health characteristics? A research
article reported the accompanying summary statistics on daily calorie intake (in calories per day) for a
representative sample of teens who do not typically eat fast food and a representative sample of teens who
do eat fast food.
Sample Sample Size Sample Mean Sample Standard Deviation
Do not eat fast food 665 2,253 1,518
Eat fast food 412 2,638 1,137
Is there convincing evidence that the mean calorie intake for teens who typically eat fast food is greater
than the mean intake for those who don't by more than 150 calories per day? (Test the relevant hypotheses
using a significance level of alpha = 0.05. Use 𝜇1 for teens who typically eat fast food and 𝜇2 for teens who
don't typically eat fast food.)
a) State the appropriate null and alternative hypotheses
b) Find the test statistic
c) Use technology to find the P-value.
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a) The appropriate null and alternative hypotheses are:
Null hypothesis (H0): μ1 - μ2 ≤ 150 (The mean calorie intake for teens who typically eat fast food is not greater than the mean intake for those who don't by more than 150 calories per day)
Alternative hypothesis (Ha): μ1 - μ2 > 150 (The mean calorie intake for teens who typically eat fast food is greater than the mean intake for those who don't by more than 150 calories per day)
b) The test statistic is calculated using the formula:
t = (x̄1 - x̄2 - d) / √(s1^2/n1 + s2^2/n2)
Where:
x̄1 and x̄2 are the sample means
s1 and s2 are the sample standard deviations
n1 and n2 are the sample sizes
d is the difference in means specified in the null hypothesis (150 in this case)
In this case:
x̄1 = 2,638
x̄2 = 2,253
s1 = 1,137
s2 = 1,518
n1 = 412
n2 = 665
d = 150
Plugging in the values:
t = (2,638 - 2,253 - 150) / √((1,137^2/412) + (1,518^2/665))
c) To find the P-value, we need to use technology such as statistical software or a calculator. By plugging in the values and using the appropriate t-distribution, the P-value can be determined.
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