Distribution Needed for Hypothesis Testing Quiz

PDF page 394: Earlier, we discussed sampling distributions. Particular distributions are associated with hypothesis testing.We will perform hypotheses tests of a population mean using a normal distribution or a Student's t-distribution. (Remember, use a Student's t-distribution when the population standard deviation is unknown and the sample size is small, where small is considered to be less than 30 observations.) We perform tests of a population proportion using a normal distribution when we can assume that the distribution is normally distributed. We consider this to be true if the sample proportion, p' , times the sample size is greater than 5 and 1- p' times the sample size is also greater then 5. This is the same rule of thumb we used when

PDF page 394: Earlier, we discussed sampling distributions. Particular distributions are associated with hypothesis testing.We will perform hypotheses tests of a population mean using a normal distribution or a Student's t-distribution. (Remember, use a Student's t-distribution when the population standard deviation is unknown and the sample size is small, where small is considered to be less than 30 observations.) We perform tests of a population proportion using a normal distribution when we can assume that the distribution is normally distributed. We consider this to be true if the sample proportion, p' , times the sample size is greater than 5 and 1- p' times the sample size is also greater then 5. This is the same rule of thumb we used when

PDF page 395: 0 Zc = σ / n The standardizing formula can not be solved as it is because we do not have μ, the population mean. However, if we substitute in the hypothesized value of the mean, μ0 in the formula as above, we can compute a Z value. This is the test statistic for a test of hypothesis for a mean and is presented in Figure 9.3. We interpret this Z value as the associated – probability that a sample with a sample mean of X could have come from a distribution with a population mean of H0 and we call this Z value Zc for “calculated”. Figure 9.3 and Figure 9.4 show this process.

PDF page 395: 0 Zc = σ / n The standardizing formula can not be solved as it is because we do not have μ, the population mean. However, if we substitute in the hypothesized value of the mean, μ0 in the formula as above, we can compute a Z value. This is the test statistic for a test of hypothesis for a mean and is presented in Figure 9.3. We interpret this Z value as the associated – probability that a sample with a sample mean of X could have come from a distribution with a population mean of H0 and we call this Z value Zc for “calculated”. Figure 9.3 and Figure 9.4 show this process.

PDF page 395: 1 3 – distribution of H0. Notice that the horizontal axis in the top panel is labeled X 's. This is the same theoretical distribution of – X 's, the sampling distribution, that the Central Limit Theorem tells us is normally distributed. This is why we can draw it with this shape. The horizontal axis of the bottom panel is labeled Z and is the standard normal distribution. Z α and -Z α , 2 2 called the critical values, are marked on the bottom panel as the Z values associated with the probability the analyst has set as the level of significance in the test, (α). The probabilities in the tails of both panels are, therefore, the same. – Notice that for each X there is an associated Zc, called the calculated Z, that comes from solving the equation above. This calculated Z is nothing more than the number of standard deviations that the hypothesized mean is from the sample mean.

PDF page 395: 1 3 – distribution of H0. Notice that the horizontal axis in the top panel is labeled X 's. This is the same theoretical distribution of – X 's, the sampling distribution, that the Central Limit Theorem tells us is normally distributed. This is why we can draw it with this shape. The horizontal axis of the bottom panel is labeled Z and is the standard normal distribution. Z α and -Z α , 2 2 called the critical values, are marked on the bottom panel as the Z values associated with the probability the analyst has set as the level of significance in the test, (α). The probabilities in the tails of both panels are, therefore, the same. – Notice that for each X there is an associated Zc, called the calculated Z, that comes from solving the equation above. This calculated Z is nothing more than the number of standard deviations that the hypothesized mean is from the sample mean.

PDF page 395: 1 3 – distribution of H0. Notice that the horizontal axis in the top panel is labeled X 's. This is the same theoretical distribution of – X 's, the sampling distribution, that the Central Limit Theorem tells us is normally distributed. This is why we can draw it with this shape. The horizontal axis of the bottom panel is labeled Z and is the standard normal distribution. Z α and -Z α , 2 2 called the critical values, are marked on the bottom panel as the Z values associated with the probability the analyst has set as the level of significance in the test, (α). The probabilities in the tails of both panels are, therefore, the same. – Notice that for each X there is an associated Zc, called the calculated Z, that comes from solving the equation above. This calculated Z is nothing more than the number of standard deviations that the hypothesized mean is from the sample mean.

PDF page 396: If the sample mean falls "too many" standard deviations from the hypothesized mean we conclude that the sample mean could not have come from the distribution with the hypothesized mean, given our pre-set required level of significance. _ _ It could have come from H0, but it is deemed just too unlikely. In Figure 9.3 both X and X are in the tails of the 1 3 distribution. They are deemed "too far" from the hypothesized value of the mean given the chosen level of alpha. If in fact this sample mean it did come from H0, but from in the tail, we have made a Type I error: we have rejected a good null. Our only real comfort is that we know the probability of making such an error, α, and we can control the size of α. _

PDF page 396: If the sample mean falls "too many" standard deviations from the hypothesized mean we conclude that the sample mean could not have come from the distribution with the hypothesized mean, given our pre-set required level of significance. _ _ It could have come from H0, but it is deemed just too unlikely. In Figure 9.3 both X and X are in the tails of the 1 3 distribution. They are deemed "too far" from the hypothesized value of the mean given the chosen level of alpha. If in fact this sample mean it did come from H0, but from in the tail, we have made a Type I error: we have rejected a good null. Our only real comfort is that we know the probability of making such an error, α, and we can control the size of α. _

PDF page 396: Figure 9.4 shows the third possibility for the location of the sample mean, x . Here the sample mean is within the two

PDF page 396: critical values. That is, within the probability of (1-α) and we cannot reject the null hypothesis.

PDF page 396: critical values. That is, within the probability of (1-α) and we cannot reject the null hypothesis.

PDF page 397: An alternative decision rule can be developed by calculating the probability that a sample mean could be found that would give a test statistic larger than the test statistic found from the current sample data assuming that the null hypothesis is true. Here the notion of "likely" and "unlikely" is defined by the probability of drawing a sample with a mean from a population with the hypothesized mean that is either larger or smaller than that found in the sample data. Simply stated, the p-value approach compares the desired significance level, α, to the p-value which is the probability of drawing a sample mean further from the hypothesized value than the actual sample mean. A large p-value calculated from the data indicates that we should not reject the null hypothesis. The smaller the p-value, the more unlikely the outcome, and the stronger the evidence is against the null hypothesis. We would reject the null hypothesis if the evidence is strongly against it. The relationship between the decision rule of comparing the calculated test statistics, Zc, and the Critical Value, Zα , and using the p-value can be seen in Figure 9.5.