Some bottles of bubble bath have gone past their expiration date. The employees want to know if their expired bubble bath produces a significantly different amount of bubbles. They know that a fresh bottle of bubble bath produces a 124 cubic inch (in3) of bubbles with a SD of 8 (in3). After testing 16 bottles, they find the average expired bottle produces 130 (in3) of bubble. What is the SEM?
The Correct Answer and Explanation is :
To calculate the Standard Error of the Mean (SEM), we can use the formula:
[
\text{SEM} = \frac{\text{SD}}{\sqrt{n}}
]
Where:
- SD is the standard deviation (8 in³ for fresh bubble bath),
- n is the sample size (16 bottles of expired bubble bath).
Now, plugging the numbers into the formula:
[
\text{SEM} = \frac{8}{\sqrt{16}} = \frac{8}{4} = 2
]
So, the SEM is 2 cubic inches (in³).
Explanation:
The Standard Error of the Mean (SEM) is a measure of how much the sample mean (the average of the 16 expired bottles in this case) is expected to vary from the true population mean. Essentially, SEM quantifies the variability of the sample mean due to random sampling. A smaller SEM indicates that the sample mean is a more precise estimate of the population mean.
In this problem, we are comparing the average bubble output of 16 expired bottles to a population mean (fresh bubble bath bottles) that has a mean of 124 in³ and a standard deviation (SD) of 8 in³. The SEM helps to determine whether the observed difference in the sample mean (130 in³) is statistically significant compared to the population mean.
In this case, the SEM of 2 cubic inches tells us that, for a sample of 16 expired bubble bath bottles, the sample mean is likely to vary by about 2 in³ from the true population mean if we were to repeat this sampling process. This helps in determining if the observed difference (6 in³) between the expired and fresh bubble bath is larger than what we would expect due to random chance.
To test if the observed difference is significant, further hypothesis testing (e.g., a t-test) could be performed using the SEM to assess whether the sample mean of 130 in³ is significantly different from the population mean of 124 in³.