The amount of water consumed each day by a healthy adult...

The amount of water consumed each day by a healthy adult follows a normal distribution with a mean of 1.4 liters. A health campaign promotes the consumption of at least 2.0 liters per day. A sample of 10 adults after the campaign shows the following consumption in liters: $$\begin{array}{llllllll}1.5 & 1.6 & 1.5 & 1.4 & 1.9 & 1.4 & 1.3 & 1.9 & 1.8 & 1.7\end{array}$$ At the .01 significance level, can we conclude that water consumption has increased? Calculate and interpret the $p$ -value.

The amount of water consumed each day by a healthy adult follows a normal distribution with a mean of 1.4 liters. A health campaign promotes the consumption of at least 2.0 liters per day. A sample of 10 adults after the campaign shows the following consumption in liters:
$$\begin{array}{llllllll}1.5 & 1.6 & 1.5 & 1.4 & 1.9 & 1.4 & 1.3 & 1.9 & 1.8 & 1.7\end{array}$$
At the .01 significance level, can we conclude that water consumption has increased? Calculate and interpret the $p$ -value.

Key Concepts

One-tailed Test

A one-tailed test is a statistical test where the region of rejection is entirely in one tail of the probability distribution. This type of test is used when the alternative hypothesis specifies a direction of the effect, such as an increase or decrease.

Normal Distribution

A normal distribution is a continuous probability distribution that is symmetric about its mean, with data near the mean more frequent in occurrence than data far from the mean. It is characterized by parameters such as the mean and standard deviation, and is foundational in statistics due to its properties and the Central Limit Theorem.

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence in a sample of data to infer that a certain condition holds for the entire population. This process involves setting up null and alternative hypotheses and comparing the test statistic to a critical value to make a decision.

Significance Level

The significance level, often denoted by alpha (?), is the threshold used in hypothesis testing to determine whether the observed data is sufficiently unlikely under the null hypothesis. It represents the probability of rejecting the null hypothesis when it is actually true, with common values being 0.05 or 0.01.

p-value

The p-value is the probability, under the assumption of the null hypothesis, of obtaining a test statistic at least as extreme as the one observed. It serves as a measure of the strength of evidence against the null hypothesis, with lower values indicating stronger evidence to reject the null.

t-test

A t-test is a type of inferential statistic used to determine if there is a significant difference between the means of a sample and a population when the population standard deviation is unknown and the sample size is small. It relies on the t-distribution, which adjusts for the increased uncertainty.

Transcript

00:01 All right, so we're looking at the amount of water consumed each day by a healthy adult, which is normally distributed, and the mean is 1 .4.

00:14 And there's a health campaign that's promoting drinking at least two liters a day.

00:21 And a sample of 10 adults showed the following consumption right here.

00:29 So we want to conclude at this point in one significance level that the water consumption has increased from 1 .4.

00:40 So h0 is going to be that the mean is less than or equal to 1 .4.

00:51 And h1, the alternative is that new is greater than 1 .4.

01:02 Increased at the means greater than 1 .4.

01:05 That's already.

01:06 I would like to double check my size, make sure i have those correct.

01:08 And i've done it correct.

01:09 Good for me.

01:10 So we need to find the average of those values.

01:13 Well mean that the average is the function values and then we're going to do the sample standard deviation good in the critical value we need to state our decision our decision rule so we want to know let's see there's 10 adults so nine degrees of freedom at the point -01 significance level one -tail test 0 .01 nine degrees of freedom is going to be in this column right there 2 .1 that's critical value so we know is the t value greater than 2 .821? we can figure it out with our mean here minus the population mean divided by this sample senior deviation divided by the square root of the sample size 2 .927 so it's just greater than the critical value but still it follows our rule so we will reject null and we can say that the the water consumption has increased.

02:27 And we're going to interpret the p value or calculate and interpret the p value.

02:30 So 2 .92 is what you need to remember.

02:36 And nine degrees of freedom, 2 .92.

02:40 It's a one -tailed test...

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