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

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

Key Concepts

Test Statistic

The test statistic is a standardized value that is computed from sample data during a hypothesis test. It measures the distance between the observed sample statistic and the value specified under the null hypothesis, expressed in units of the estimated standard error. This value is then compared to a theoretical distribution to determine significance.

Hypothesis Testing

This is a statistical method used to make decisions about population parameters based on sample data. In hypothesis testing, a null hypothesis (which usually represents the status quo or a claim of equality) is tested against an alternative hypothesis that represents a deviation from the null. Conclusions are drawn using sample statistics and the likelihood of the observed data under the assumption that the null hypothesis is true.

t-Test

A t-test is used when the population standard deviation is unknown and the sample size is moderately small. It involves comparing the sample mean to a known value or the means of two samples. The t-distribution, which has heavier tails than the normal distribution, is used to account for the extra uncertainty introduced by estimating the standard deviation from the sample.

Degrees of Freedom

Degrees of freedom refer to the number of independent values that are free to vary in the calculation of a statistic. In the context of a t-test for a single sample, the degrees of freedom are typically the sample size minus one. This parameter is crucial since it affects the shape of the t-distribution used in the test.

p-Value

The p-value is the probability, under the null hypothesis, of obtaining a test statistic as extreme as or more extreme than the one actually observed. It quantifies the evidence against the null hypothesis. A low p-value indicates that the observed data are unlikely under the null hypothesis, leading to its rejection in favor of the alternative hypothesis.

Significance Level

The significance level, denoted by alpha (?), is a threshold set by the researcher before conducting the test. It represents the probability of rejecting the null hypothesis when it is actually true (a Type I error). By comparing the p-value to the significance level, one can decide whether to reject or fail to reject the null hypothesis.

Transcript

00:01 So the waste loss program claims that the mean weight loss in a couple of weeks is 10 pounds or more.

00:09 And alternately, we want to find out if it's actually less than 10 pounds.

00:14 And our test is to use a 5 % significance level.

00:19 And they obtained a sample from 50 people and found that the mean weight loss of those 50 people was 9 pounds.

00:27 And the standard deviation of those people was 2 .8 pounds.

00:31 And so if we have a sample size, that is 50, and that is greater than or equals to 30, so although a t test is appropriate to use for this, we can use a z value since that sample size is large enough.

00:46 It's over 30.

00:48 So we're doing a one -tail test with a lower tail, and let's find what that z value is that has 0 .05, that critical value.

00:57 And the value that is negative because it's in the lower tail and we would reject our no if we're over here and we would fail to reject our no if our test statistic is over this way.

01:09 And so our value that has 5 % in the upper tail is going to end up being a negative 1 .645.

01:20 So that's the value we would test it again.

01:22 So if we are less than or equal to that value that will be evidence that we will reject.

01:27 Our no.

01:29 So let's find that test statistic.

01:32 And it's a z value.

01:34 We'll take our 9 minus our 10, and then our sample standard deviation divided by the square root of, i believe it's 50, yes, it's 50...

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