Miller (2008) examined the energy drink consumption of...

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 that a researcher selects a random sample of $n=36$ male undergraduates and a sample of $n=25$ females. On average, the males reported consuming $M=2.45$ drinks per month and females had an average of $M=1.28 .$ Assume that the overall level of consumption for college undergraduates averages $\mu=1.85$ energy drinks per month, and that the distribution of monthly consumption scores is approximately normal with a standard deviation of $\sigma=1.2.$ a. Does this sample of males support the conclusion that males consume significantly more energy drinks than the overall population average? Use a one-tailed test with $\alpha=.01$. b. Does the sample of females support the conclusion that females consume significantly fewer energy drinks than the overall population average? Again, use a one-tailed test with $\alpha=.01$.

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 that a researcher selects a random sample of $n=36$ male undergraduates and a sample of $n=25$ females. On average, the males reported consuming $M=2.45$ drinks per month and females had an average of $M=1.28 .$ Assume that the overall level of consumption for college undergraduates averages $\mu=1.85$ energy drinks per month, and that the distribution of monthly consumption scores is approximately normal with a standard deviation of $\sigma=1.2.$
a. Does this sample of males support the conclusion that males consume significantly more energy drinks than the overall population average? Use a one-tailed test with $\alpha=.01$.
b. Does the sample of females support the conclusion that females consume significantly fewer energy drinks than the overall population average? Again, use a one-tailed test with $\alpha=.01$.

Key Concepts

Z-Test

The Z-test is a statistical test used when the population standard deviation is known and the sample size is sufficiently large or the population distribution is normal. It involves comparing the sample mean to the population mean using the standard error, and then converting this difference into a standardized z-score to assess statistical significance.

Test Statistic and Decision Rule

The test statistic, such as the z-score in a Z-test, quantifies the difference between the sample data and the population parameter in units of standard error. The decision rule involves comparing the computed test statistic to a critical value from a statistical distribution (e.g., the Z-distribution) corresponding to the chosen significance level, to determine whether to reject or fail to reject the null hypothesis.

One-Tailed Test

A one-tailed test is used when the research question or hypothesis is directional, meaning that the effect is expected to occur in one specific direction (either greater than or less than a certain value). This test increases the statistical power to detect an effect in that specific direction, but it does not allow for detecting effects in the opposite direction.

Standard Error of the Mean

The standard error of the mean measures how much the sample mean is expected to vary from the true population mean. It is calculated by dividing the population standard deviation by the square root of the sample size, and it provides a basis for estimating the precision of the sample mean as an estimate of the population mean.

Null and Alternative Hypotheses

The null hypothesis (H0) typically represents the status quo or a statement of no effect, while the alternative hypothesis (H1) represents the outcome that the researcher is testing for. They are mutually exclusive and collectively exhaustive, allowing researchers to determine if there is strong evidence to reject the null in favor of the alternative.

Hypothesis Testing

Hypothesis testing is a systematic method used to decide whether there is enough statistical evidence in favor of a particular belief about a population parameter. It involves comparing observed sample data against what is expected under a null hypothesis to determine if any observed differences are statistically significant.

Significance Level (?)

The significance level, often denoted by ?, is the threshold for rejecting the null hypothesis. It represents the probability of committing a Type I error, which is the error of rejecting a true null hypothesis. The chosen ? level, such as 0.01, sets the standard for how extreme the data must be before concluding that the observed effect is statistically significant.

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