Hypothesis Tests 3

In a significance test, hypotheses are statements made about the population parameters. These parameters are characteristics or measurements of the entire population being studied. Therefore, hypotheses are always stated in terms of these population parameters, rather than sample statistics or any other variables. This is because the goal of a significance test is to make inferences about the population based on the sample data, and to determine if the observed results are statistically significant or due to chance.

A t-test is used when the population standard deviation for a mean is unknown because it allows us to make inferences about the population mean using a sample mean. When the population standard deviation is unknown, the t-test uses the sample standard deviation to estimate it, providing a more accurate measure of uncertainty. This is in contrast to the z-test, which requires knowledge of the population standard deviation. Therefore, the statement that a t-test is used when the population standard deviation for a mean is unknown is true.

The critical value (z*) is used to determine the margin of error in a confidence interval. In this case, a 99% confidence interval is being calculated for a sample mean with a known population standard deviation. The critical value for a 99% confidence level is 2.576. Therefore, the given statement is true.

If the p-value for a two-sided test is 0.065, then the p-value for a one-sided test would be half of that, assuming the effect is in the predicted direction, because the two-sided test allocates half of the alpha level to each tail of the distribution. So, for a one-sided test, the p-value would be 0.065 / 2 = 0.0325.

Even though the p-value for a one-sided test would be lower than the two-sided test, it would not be significant at the 1% level (0.01), as 0.0325 is greater than 0.01. Therefore, the statement is true.

If the results of a hypothesis test are significant at the 1% level, it means that the probability of obtaining the observed results by chance is very low (less than 1%). Since the 5% level is less strict than the 1% level, if the results are significant at the 1% level, they will also be significant at the 5% level. This is because if the results are unlikely to occur by chance at a 1% level, they will also be unlikely to occur at a 5% level. Therefore, the statement is true.

The statement is false because the p-value is not the probability that Ho (null hypothesis) is true. The p-value is the probability of obtaining the observed data or more extreme results, assuming that the null hypothesis is true. It is used to determine the statistical significance of the results and make decisions regarding the rejection or acceptance of the null hypothesis. The p-value does not provide direct information about the probability of the null hypothesis being true.

When a p-value is high, it means that there is not enough evidence to reject the null hypothesis. This implies that the observed data is consistent with the null hypothesis being true. In other words, a high p-value suggests that the results are likely due to chance and not a result of the alternative hypothesis. Therefore, the statement "there is strong evidence against the null hypothesis" is incorrect.

The given confidence interval (8.67, 11.26) suggests that the population mean is likely to fall within this range. Therefore, it is unlikely that a 2-sided test for the hypothesis would find evidence to reject the null hypothesis (Ho) at the 5% level. The correct answer is False.

When a sample statistic is close to the believed population parameter, it suggests that the null hypothesis is likely to be true. In this case, the p-value for a significance test will typically be high, indicating that the observed data is likely to occur by chance alone and not due to a significant effect. Therefore, the correct answer is false.