Do you drink the recommended amount of water each day? Most Americans don't! On average, Americans drink 5.9 eight-oz servings of water a day. A sample of 50 education professionals was randomly selected and their water consumption for a 24-hour period was monitored; the mean amount consumed was 49.1 oz. Assuming the amount of water consumed daily by adults is normally distributed and the standard deviation is 10.9 oz, is there sufficient evidence to show that education professionals consume, on average, more water daily than the national average? Use á = .05.
(a) Find z. (Give your answer correct to two decimal places.)
(ii) Find the p-value. (Give your answer correct to four decimal places.)
To determine if education professionals consume more water daily than the national average, we can calculate the z-score and the p-value.
(a) Find z:
The z-score formula is: z = (x - μ) / σ
Where:
x = sample mean (49.1 oz)
μ = population mean (5.9 * 8 oz = 47.2 oz)
σ = population standard deviation (10.9 oz)
Plugging in the values:
z = (49.1 - 47.2) / 10.9
Calculating the z-score:
z ≈ 0.174
The z-score is approximately 0.17.
(ii) Find the p-value:
The p-value is the probability of getting a sample mean as extreme as 49.1 oz, assuming the null hypothesis is true (i.e., the population mean is equal to the national average).
To find the p-value, we need to use a z-table or a statistical software. Since we already have the z-score, we can use the standard normal distribution table (z-table).
1. Look for the z-score in the z-table. In this case, we find 0.17.
2. The p-value is the area to the right of the z-score in the table. Since we're looking for the probability of getting a more extreme value, we need to find the area to the right (which will be subtracted from 1).
The p-value is approximately 1 - 0.5662 ≈ 0.4338.
Therefore, the p-value is approximately 0.4338.
Conclusion:
With a p-value of 0.4338, and using a significance level of 0.05, we do not have sufficient evidence to show that education professionals consume more water daily than the national average.
To find the z-score, we will use the formula:
z = (x - μ) / σ
Where:
x = sample mean = 49.1 oz
μ = population mean = 5.9 eight-oz servings * 8 oz/serving = 47.2 oz
σ = standard deviation = 10.9 oz
Substituting the values into the formula:
z = (49.1 - 47.2) / 10.9
z ≈ 0.174
To find the p-value, we need to use the z-table. Since we are testing whether education professionals consume more water daily than the national average, we are interested in the upper tail of the distribution.
Using the z-table, we can find the area to the right of z = 0.174, which corresponds to the p-value.
The p-value is approximately 0.4301. (Round to four decimal places.)
Therefore, the p-value is 0.4301.