Question

Water! A blogger claims that U.S. adults drink an average of five $8-$ounce glasses (that’s $40$ ounces) of water per day. Researchers wonder if this claim is true, so they ask a random sample of $24$ U.S. adults about their daily water intake. A graph of the data shows a roughly symmetric shape with no outliers.

a. State an appropriate pair of hypotheses for a significance test in this setting. Be sure to define the parameter of interest.

b. Check conditions for performing the test in part (a).

c. The $90\%$ confidence interval for the mean daily water intake is $30.35$ to $36.92$ ounces. Based on this interval, what conclusion would you make for a test of the hypotheses in part (a) at the $10\%$ significance level?

d. Do we have convincing evidence that the amount of water U.S. children drink per day differs from $40$ ounces? Justify your answer.

Short answer

Part (a)$$\begin{gathered} H_0:\mu =40 \\ {H}_1:\mu \ne 40 \end{gathered}$$

Part (b) All conditions are satisfied.

Part (c) There is convincing proof that the mean daily water intake is different from $40$ounces.

Part (d) No.

Step-by-step solution

Part (a) Step 1: Given information

Claim is that the mean is $40$ ounces.

Part (a) Step 2: Explanation

The null hypothesis asserts that the population value is the same as the claim value:

$H_0:\mu =40$

Either the null hypothesis or the alternative hypothesis is the assertion. The null hypothesis asserts that the population means is the same as the claim value. If the claim is the null hypothesis, the alternative hypothesis statement is the polar opposite of the claim.

$H_1:\mu \ne 40$

$\mu$is the mean daily water intake of U.S. adults.

Part (b) Step 1: Explanation

The three conditions are Random, independent, and Normal/ Large sample.

Random: Because the sample is a random sample, I'm satisfied.

Independent: Because the sample of $24$ individuals in the United States represents less than $10\%$ of the total adult population in the United States, I am happy.

Normal/Large sample: Satisfied since the data graph reveals no outliers or skewed distribution. Because all of the prerequisites are met, a hypothesis test for the population mean is appropriate.

Part (c) Step 1: Given information

$90\%$ confidence interval:

$(30.35,36.92)$

Part (c) Step 2: Explanation

A hypothesis test with a significance level of $100\%-90\%=10\%$ is associated with a $90\%$ confidence interval.

It is noted that the confidence interval does not include $40$ implying that the mean daily water consumption is unlikely to be $40$ ounces, and so there is convincing proof that the mean daily water intake is not $40$ ounces.

Part (d) Step 1: Explanation

Because the data is based on adults in the United States, it is impossible to conclude that children in the United States have a different daily water intake than adults.

It signifies that we don't have enough data to believe that the average daily water intake of American youngsters is less than $40$ ounces.