Miller (2008) examined the energy drink consumption of college undergraduates and found that males use energy drinks significantly more often than females. To further investigate this phenomenon, suppose a researcher selects a random sample of n=36 male undergraduates and a sample of n=25 females. On average, the males reported consuming M=2.45 drink per month and females had an average of M=1.28. Assume that the overall level of consumption for college undergraduates averages population mean=1.85 energy drinks per month, and that the distribution of montly consumption scores is approximately normal with a standard deviation of 1.2.
a) Did this sample of males consume significantly more energy drinks than the overall population average? Use a one-tailed test with a a=.01
b) Did this sample of females consume significantly fewer energy drinks than the overall population average? Use a one-tailed test with a a=.01
Z = (mean1 - mean2)/standard error (SE) of difference between means
SEdiff = √(SEmean1^2 + SEmean2^2)
SEm = SD/√n
If only one SD is provided, you can use just that to determine SEdiff.
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability for each Z score.
a) To determine if the sample of males consumed significantly more energy drinks than the overall population average, we can conduct a one-sample t-test.
Step 1: State the hypotheses
- Null hypothesis (H0): The mean consumption of energy drinks for males is equal to the overall population mean (µ = 1.85).
- Alternative hypothesis (Ha): The mean consumption of energy drinks for males is greater than the overall population mean (µ > 1.85).
Step 2: Set the significance level
The significance level, α, is given as 0.01 or 1%.
Step 3: Compute the test statistic
The formula for the t-test statistic in this case is: t = (X - µ) / (s / sqrt(n))
Where:
- X = sample mean
- µ = population mean
- s = sample standard deviation
- n = sample size
Using the given information, we have:
- X = 2.45 (sample mean)
- µ = 1.85 (population mean)
- s = 1.2 (population standard deviation)
- n = 36 (sample size)
Substituting these values into the formula gives:
t = (2.45 - 1.85) / (1.2 / sqrt(36))
Step 4: Determine the critical value
Since the alternative hypothesis is one-tailed and we are testing if the mean is greater, we need to find the critical value for a upper one-tailed test at α = 0.01. Looking up the critical value in the t-distribution table gives approximately 2.423.
Step 5: Make a decision
If the test statistic (t) is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Calculating the test statistic:
t = (2.45 - 1.85) / (1.2 / sqrt(36))
t ≈ 1.732
Since 1.732 is not greater than 2.423, we fail to reject the null hypothesis.
Step 6: State the conclusion
There is not enough evidence to suggest that the sample of males consumed significantly more energy drinks than the overall population average at α = 0.01.
b) To determine if the sample of females consumed significantly fewer energy drinks than the overall population average, we can conduct a one-sample t-test.
Step 1: State the hypotheses
- Null hypothesis (H0): The mean consumption of energy drinks for females is equal to the overall population mean (µ = 1.85).
- Alternative hypothesis (Ha): The mean consumption of energy drinks for females is less than the overall population mean (µ < 1.85).
Step 2: Set the significance level
The significance level, α, is given as 0.01 or 1%.
Step 3: Compute the test statistic
Using the given information:
- X = 1.28 (sample mean)
- µ = 1.85 (population mean)
- s = 1.2 (population standard deviation)
- n = 25 (sample size)
The formula for the t-test statistic in this case is: t = (X - µ) / (s / sqrt(n))
Calculating the test statistic:
t = (1.28 - 1.85) / (1.2 / sqrt(25))
t ≈ -2.25
Step 4: Determine the critical value
Since the alternative hypothesis is one-tailed and we are testing if the mean is less, we need to find the critical value for a lower one-tailed test at α = 0.01. Looking up the critical value in the t-distribution table gives approximately -2.485.
Step 5: Make a decision
If the test statistic (t) is less than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Since -2.25 is not less than -2.485, we fail to reject the null hypothesis.
Step 6: State the conclusion
There is not enough evidence to suggest that the sample of females consumed significantly fewer energy drinks than the overall population average at α = 0.01.
To answer these questions, we need to conduct a hypothesis test. Let's go through the steps for each question:
a) Did this sample of males consume significantly more energy drinks than the overall population average? Use a one-tailed test with α=0.01.
Step 1: State the null hypothesis (H0) and alternative hypothesis (Ha):
H0: The average energy drink consumption for males is not significantly different from the overall population average (μ = 1.85).
Ha: The average energy drink consumption for males is significantly higher than the overall population average (μ > 1.85).
Step 2: Set the criteria for decision-making (α level):
α = 0.01 (given in the question).
Step 3: Calculate the test statistic:
We need to use the z-test since the sample size is relatively large, and we know the population standard deviation.
z = (sample mean - population mean) / (population standard deviation / √sample size)
z = (2.45 - 1.85) / (1.2 / √36)
Step 4: Determine the critical value:
Since this is a one-tailed test, we need to look up the critical value for a one-tailed test with α = 0.01, which corresponds to z = 2.33 (approximately).
Step 5: Make a decision:
Compare the test statistic with the critical value.
If the test statistic is greater than the critical value (z > 2.33), reject the null hypothesis.
If the test statistic is less than or equal to the critical value (z ≤ 2.33), fail to reject the null hypothesis.
Step 6: State the conclusion:
If we reject the null hypothesis, we can conclude that the sample of males consumed significantly more energy drinks than the overall population average. If we fail to reject the null hypothesis, we do not have enough evidence to conclude that the sample of males consumed significantly more energy drinks than the overall population average.
b) Did this sample of females consume significantly fewer energy drinks than the overall population average? Use a one-tailed test with α=0.01.
Step 1: State the null hypothesis (H0) and alternative hypothesis (Ha):
H0: The average energy drink consumption for females is not significantly different from the overall population average (μ = 1.85).
Ha: The average energy drink consumption for females is significantly lower than the overall population average (μ < 1.85).
Step 2: Set the criteria for decision-making (α level):
α = 0.01 (given in the question).
Step 3: Calculate the test statistic:
We need to use the z-test since the sample size is relatively large, and we know the population standard deviation.
z = (sample mean - population mean) / (population standard deviation / √sample size)
z = (1.28 - 1.85) / (1.2 / √25)
Step 4: Determine the critical value:
Since this is a one-tailed test, we need to look up the critical value for a one-tailed test with α = 0.01, which corresponds to z = -2.33 (approximately).
Step 5: Make a decision:
Compare the test statistic with the critical value.
If the test statistic is less than the critical value (z < -2.33), reject the null hypothesis.
If the test statistic is greater than or equal to the critical value (z ≥ -2.33), fail to reject the null hypothesis.
Step 6: State the conclusion:
If we reject the null hypothesis, we can conclude that the sample of females consumed significantly fewer energy drinks than the overall population average. If we fail to reject the null hypothesis, we do not have enough evidence to conclude that the sample of females consumed significantly fewer energy drinks than the overall population average.