A coffee manufacturer is interested in whether the mean...

A coffee manufacturer is interested in whether the mean daily consumption of regular coffee drinkers is less than that of decaffeinated-coffee drinkers. A random sample of 50 regular-coffee drinkers showed a mean of 4.35 cups per day, with a standard deviation of 1.20 cups per day. A sample of 40 decaffeinated-coffee drinkers showed a mean of 5.84 cups per day, with a standard deviation of 1.36 cups per day. Use the .01 significance level. Compute the $p$ -value.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about population parameters based on sample data. It involves setting up a null hypothesis (a statement of no effect or difference) and an alternative hypothesis (a statement that contradicts the null), then using sample data to decide whether there is sufficient evidence to reject the null hypothesis. This process helps researchers determine if observed differences are statistically significant or might have occurred by chance.

Two-Sample t-Test

A two-sample t-test is used to determine if there is a significant difference between the means of two independent groups. It compares the means by taking into account the variability in each group as well as the sample sizes. This test can either assume equal variances between the populations or use a variation like Welch's t-test when the variances are unequal.

One-Tailed Test

A one-tailed test is a type of hypothesis test in which the alternative hypothesis specifies a direction of the difference (either greater than or less than) rather than simply not equal. This approach is used when the research question or objective predicts that the effect will occur in a specified direction, and it concentrates all the significance level in one tail of the distribution.

p-value

The p-value is the probability of obtaining test results at least as extreme as the results actually observed, under the assumption that the null hypothesis is true. It is a measure used to quantify the evidence against the null hypothesis; a small p-value indicates that the observed data are unlikely under the null hypothesis, leading to its potential rejection.

Significance Level

The significance level, often denoted as alpha (?), is the threshold set by the researcher that determines when to reject the null hypothesis. It represents the probability of committing a Type I error, which is rejecting the null hypothesis when it is actually true. A lower significance level means stricter criteria for rejecting the null hypothesis.

Transcript

00:01 So at a 1 % significance level, we want to see do regular coffee drinkers, and we'll let regular be number one, and decaffeinated, decaffeinated be number two.

00:16 And we want to find out, would the mean number of cups being consumed be equal? and they alternately think that the decaffeinated actually drink more or that the caffeinated, the caffeinated drink less than the decaffeinated.

00:33 So let's see if we have evidence for that.

00:35 And so we have our data.

00:40 We have a first sample size is 50 for the caffeinated people, and i'm a caffeinated person.

00:48 And that mean is 4 .35 cups per day.

00:53 And the standard deviation of that sample came out to be 1 .2 cups.

00:58 And for the decaffeinated coffee, we had a mean of 5 .84 with a sample standard deviation of 1 .36 and a sample size of 40.

01:13 Now, typically, if we don't know the population standard deviation, we may have to use a t value.

01:20 But since both our sample sizes are greater than or equal to 30, we can use a z value to estimate this probability.

01:29 So let's look over here and let's just draw where our critical values will be.

01:34 And we're going to put all 1 % because of it being a one -tail test down in this lower tail.

01:40 And this is where we will be rejecting the null if our test statistic is down here.

01:45 And we will fail to reject if it's up here.

01:48 And this z value, let me quick look, because i don't know what 1 % value is, that is negative 2 .36.

01:58 That's not one i have memorized.

01:59 So let's calculate that test statistic, that z value...

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