A new weight-watching company, Weight Reducers...

A new weight-watching company, Weight Reducers International, advertises that those who join will lose an average of 10 pounds after the first two weeks. The standard deviation is 2.8 pounds. A random sample of 50 people who joined the weight reduction program revealed a mean loss of 9 pounds. At the .05 level of significance, can we conclude that those joining Weight Reducers will lose less than 10 pounds? Determine the $p$ -value.

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to determine whether there is enough evidence in a sample of data to infer that a certain condition holds true for the entire population. It involves setting up a null hypothesis (which represents the status quo) and an alternative hypothesis, and then using sample data to decide whether to reject the null hypothesis.

One-Sample z-Test

A one-sample z-test is used when comparing the sample mean to a known population mean while the population standard deviation is known. This test calculates a z-score that measures the number of standard errors the sample mean is away from the hypothesized population mean, and is particularly useful when the sample size is large or the population data is normally distributed.

Standard Error

The standard error is the estimated standard deviation of the sampling distribution of the sample mean. It is calculated by dividing the population standard deviation by the square root of the sample size, and it quantifies the variability of the sample mean estimates from the true population mean.

Significance Level

The significance level, denoted by ?, is the threshold used to determine whether the results of a hypothesis test are statistically significant. It represents the probability of incorrectly rejecting the null hypothesis, and common values for ? are 0.05, 0.01, etc.

P-Value

The p-value is the probability of obtaining a test statistic at least as extreme as the one observed, assuming that the null hypothesis is true. A small p-value indicates that the observed data is unlikely under the null hypothesis, which may lead to its rejection in favor of the alternative hypothesis.

Transcript

00:01 All right, so a new dieting company advertises that those who join will lose an average of 10 pounds after two weeks.

00:13 Dang, that's nice, with a standard deviation of 2 .8 pounds.

00:16 So they sampled 50 people who joined the program, and they found a mean loss of 9 pounds.

00:22 So not quite a 10.

00:24 So at the 0 .05 level of significance, can we conclude that those who join this program will lose less than 10 pounds? all right, so now we need to think about what we have here.

00:34 So we have the population, 10 pounds, with a given standard deviation.

00:39 So we're going to use the z score.

00:43 And let's see.

00:45 So that means we can, let's calculate that right away.

00:49 Let's see, the mean, the sample was 9 minus the mean of the population is 10, divided by the standard deviation, 2 .8 divided by the square of the sample size, which is 50.

01:05 So that's the z score.

01:06 Now, before we do the critical value, let's make sure we know our hypothesis here.

01:11 So the null hypothesis, hang out, let's read this question again.

01:14 You've got to read this very carefully.

01:16 Can we conclude that those joining the program will lose less than 10 pounds? so we want to say, is the mean, let's see, let's see if this makes sense.

01:32 Will the mean, we're testing, is the mean less than 10 pounds? that means the null hypothesis, would be no, it's greater than equal to 10 pounds.

01:47 Right? so let's see if this makes sense with what we have here.

01:50 Can we conclude that those joining weight, this weight program will lose less than 10? will the mean be less than 10? will the sample mean be less than 10? is that an expected outcome? that means the null hypothesis would be 10 or greater because that's what this says here.

02:16 They're saying we'll lose an average of 10 pounds in the first two weeks...

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