A coffee-dispensing machine is supposed to deliver 8 ounces of liquid into each paper cup, but a consumer believes that the actual mean amount is not equal to 8 ounces. The consumer obtained a sample of 25 cups of the dispensed liquid with a sample mean of 7.8 ounces and a sample variance of 0.81 ounces. If the dispensed liquid delivered per cup is normally distributed, perform a hypothesis test using α=0.01 to determine if, on average, the coffee-dispensing machine delivers 8 ounces of liquid into each paper cup.
The researcher's null hypothesis and alternative are:
Null hypothesis (H0): The mean amount of liquid delivered by the coffee-dispensing machine is equal to 8 ounces.
Alternative hypothesis (Ha): The mean amount of liquid delivered by the coffee-dispensing machine is not equal to 8 ounces.
To perform this hypothesis test, the appropriate statistic is the t-statistic.
The test statistic value for this test can be calculated using the formula:
t = (sample mean - hypothesized mean) / (sample standard deviation / sqrt(sample size))
In this case, the sample mean is 7.8 ounces, the hypothesized mean is 8 ounces, the sample standard deviation is the square root of the sample variance (0.81 ounces), and the sample size is 25 cups.
The conclusion should be drawn by comparing the calculated t-statistic value with the critical t-value at a significance level of α=0.01. If the calculated t-statistic falls within the critical region, the null hypothesis can be rejected, indicating that the coffee-dispensing machine does not deliver 8 ounces of liquid on average. If the calculated t-statistic does not fall within the critical region, there is not enough evidence to reject the null hypothesis, suggesting that the coffee-dispensing machine delivers 8 ounces of liquid on average.