A proposed null hypothesis states that there is no difference in the water consumption of households of two districts. The sample mean difference is found to be 8.5 liters, and the standard deviation of the distribution of the difference in sample means is 4.5 liters. Which statement is true?

What statements?

112 To determine which statement is true, we need to compare the sample mean difference to the distribution of the difference in sample means, taking into account the standard deviation.

Since the null hypothesis states that there is no difference in water consumption between the households of the two districts, the expected value of the difference in means would be zero. If the sample mean difference is significantly different from zero, we can reject the null hypothesis and conclude that there is a difference in water consumption.

Now, to analyze whether the sample mean difference of 8.5 liters is significant, we need to calculate the z-score. The z-score measures how many standard deviations the sample mean difference is away from the expected value (in this case, zero).

The formula to calculate the z-score is:

z-score = (sample mean difference - expected value) / standard deviation

In this case, the sample mean difference is 8.5 liters, the expected value is 0 liters, and the standard deviation is 4.5 liters. Plugging these values into the formula, we get:

z-score = (8.5 - 0) / 4.5 = 1.89

To determine if this z-score is significant, we refer to the standard normal distribution table or use statistical software to find the corresponding p-value. The p-value is the probability of observing a sample mean difference as extreme as 8.5 liters or more, assuming the null hypothesis is true.

If the p-value is less than a predefined significance level - commonly 0.05, 0.01, or 0.1 - then we can reject the null hypothesis. If the p-value is greater than or equal to the significance level, we fail to reject the null hypothesis.

Given the information provided, we cannot determine the p-value or the significance level. Therefore, without this information, we cannot definitively state which statement is true.