The following sample of six measurements was randomly selected from a normally distributed popul

The following sample of six measurements was randomly...

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions or inferences about a population parameter based on sample data. The procedure involves formulating two competing hypotheses, the null hypothesis which represents a baseline assumption, and the alternative hypothesis which represents the claim to be tested. Data is then used to determine whether there is enough evidence to reject the null hypothesis in favor of the alternative hypothesis.

Null and Alternative Hypotheses

The null hypothesis (H?) typically asserts that there is no effect or difference, serving as the default assumption, while the alternative hypothesis (H?) represents what the researcher aims to support, indicating the presence of an effect or difference. In different tests, the alternative hypothesis can be directional (specifying a direction of the effect) or non-directional (simply indicating a difference).

One-tailed vs Two-tailed Tests

A one-tailed test is used when the alternative hypothesis specifies a direction of the effect (e.g., the mean is less than or greater than a hypothesized value), while a two-tailed test is used when the alternative hypothesis simply indicates that the parameter differs from the hypothesized value without direction. The choice between these tests affects the critical values and the interpretation of the test results.

Significance Level

The significance level, denoted by alpha (?), is the threshold probability used to decide whether to reject the null hypothesis. It represents the maximum acceptable risk of making a Type I error, which is rejecting the null hypothesis when it is actually true. Common choices for ? include 0.05 or 0.01.

Observed Significance Level (p-value)

The observed significance level, or p-value, is the probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true. It quantifies the evidence against the null hypothesis, with smaller p-values indicating stronger evidence to reject the null hypothesis. Decisions are made by comparing the p-value to the predetermined alpha level.

t-test and t-distribution

A t-test is a type of inferential statistic used to determine if there is a significant difference between the sample mean and a known population mean, especially when the sample size is small and the population standard deviation is unknown. The t-test makes use of the t-distribution, which accounts for the additional uncertainty involved in estimating the population standard deviation from the sample.

Degrees of Freedom

Degrees of freedom refer to the number of independent values in a statistical calculation that are free to vary while estimating a parameter. In the context of a t-test, the degrees of freedom, usually calculated as the sample size minus one (n-1), affect the shape of the t-distribution and are critical for determining the critical values for the test.

Transcript

00:01 For this problem, we are told that the following sample of six measurements were randomly selected from a normally distributed population.

00:08 The first thing that i'll note is that we'll find that the sample standard deviation is going to be equal to...

00:14 One moment here.

00:15 All right, just wanted to double check a value i found.

00:19 All right, so this is going to be s equals 3 .098 for the sample standard deviation.

00:26 And we'll find that the sample mean, x bar, is actually going to be equal to 3.

00:31 In part a, we are asked to test the null hypothesis that the mean of the population is 2 against the alternative hypothesis, mu, less than 2, where we are to use alpha equals 0 .05.

00:45 So our first step here, since we have a very small sample size, will be to calculate the t value, which will be equal to x bar minus mu, divided by s over the square root of n, which will then become clearly 3.

01:01 098, or excuse me, it would be 3 minus 2 rather, 3 minus 2 over 3 .098, divided by the square root of 6, and we'll get that t is going to be equal to 0 .79.

01:21 I'll note that we'll use the same t value for both of these tests.

01:28 Additionally, we'll want to, using this level of significance, and noting that we have a one -tailed test, term the critical t value, in which case we'd have that with five degrees of freedom, the critical t value is going to be negative 2 .015.

01:43 We can see that our calculated t value, let's see here, t equals 0 .79, is greater than our critical t value, but we have to keep in mind this is a one -tailed lesser or to the left sort of test, which means that this, that we are not, not in the rejection region so we fail to reject fail to reject in part b we are asked to to test the null hypothesis that the mean of the population is two against the alternative hypothesis that mu does not equal to where we are to use the same alpha level so for referencing the appropriate tables we'd want to be using the alpha over 2 value of 0 .025 and we'll find that the critical t value here would be equal to let's see 2 .5706.

02:39 2 .5706.

02:41 So our region of rejection is going to be where the absolute value of t is greater than 2 .5706, but we can see that 0 .79 is less than 2 .5706.

02:56 So we fail.

03:00 One moment here...

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