Statistics Answer Note #11

Question 1 John works at the ISU Horticulture Center during the summer. Three years ago, 85% of the center’s gardens were visited by native bees. John wants to test if this proportion (p) has changed since then. Write out this scenario's null hypothesis (H0) and alternative hypothesis (H1). Answer H0: The proportion of gardens visited by native bees at the ISU Horticulture Center is equal to 85%. H1: The proportion of gardens visited by native bees at the ISU Horticulture Center is not equal to 85%. Explanation: Null and alternative hypotheses are an integral part of hypothesis testing in statistics. The null hypothesis, denoted by H0, is a statement that assumes there is no difference or relationship between two variables of interest. It is considered as the default position or initial belief unless there is strong evidence to suggest otherwise. On the other hand, the alternative hypothesis, denoted by H1, is a statement that contradicts the null hypothesis and proposes a different relationship between the variables. In the given scenario, the null hypothesis states that the proportion of gardens visited by native bees at the ISU Horticulture Center is equal to 85%. This means that there has been no change in the proportion of gardens visited by native bees since three years ago. The alternative hypothesis, on the other hand, suggests that the proportion is not equal to 85%. This could mean that there has been an increase or decrease in the proportion of gardens visited by native bees. To test these hypotheses, John would need to collect data on the number of gardens visited by native bees during the current summer season and compare it to the proportion observed three years ago. Depending on the type of data and the research question, appropriate statistical tests such as the z-test or chi-square test can be performed to determine the significance and validity of the results. Question 2 If we reject the null hypothesis when the null hypothesis is true (unbeknownst to us), what type of error have we made? Answer When conducting a statistical hypothesis test, there is always a chance that the decision made based on the sample data may be incorrect. This can result in two types of errors Type I error and Type II error. A Type I error, also known as a false positive, occurs when the null hypothesis is rejected when it is actually true. This means that the statistical test has detected a significant difference when in reality, there is no true difference between the variables being tested. It is similar to a person being wrongly convicted of a crime they did not commit. On the other hand, a Type II error, also known as a false negative, occurs when the null hypothesis is accepted when it is actually false. This means that the statistical test has failed to detect a significant difference when in reality, there is a true difference between the variables being tested. It is similar to a person being wrongly acquitted of a crime they actually committed. In the given question, if we reject the null hypothesis when it is actually true (unbeknownst to us), we are committing a Type I error. This is because we have mistakenly concluded that there is a significant difference between the variables, when in reality there is none. To understand the concept of Type I and Type II errors further, let us look at their definitions and formulas: Type I error: - Definition: Rejecting the null hypothesis when it is actually true. - Formula: P(Type I error) = alpha level (α) Type II error: - Definition: Accepting the null hypothesis when it is actually false. - Formula: P(Type II error) = beta level (β) The alpha level (α) and beta level (β) are predetermined probabilities set by the researcher. The alpha level is the probability of rejecting the null hypothesis when it is actually true, and the beta level is the probability of accepting the null hypothesis when it is actually false. These probabilities are inversely related, meaning that as one increases, the other decreases. Question 3 If we reject a null hypothesis, should we state that we have proved the alternative hypothesis true? Please explain why or why not. Answer In the field of statistics, it is common to encounter two types of hypotheses: the null hypothesis and the alternative hypothesis. The null hypothesis is generally the default or conventional hypothesis, while the alternative hypothesis proposes an alternative explanation to the null hypothesis. The goal of hypothesis testing is to determine whether there is enough evidence to reject the null hypothesis and support the alternative hypothesis. However, it is important to understand that rejecting the null hypothesis does not necessarily mean proving the alternative hypothesis to be true. It is essential to note that the null hypothesis and alternative hypothesis are two sides of the same coin. Rejecting the null hypothesis only suggests that there is enough evidence to believe that the alternative hypothesis is more likely to be true. It does not prove the alternative hypothesis to be true with absolute certainty. Essentially, rejecting the null hypothesis simply means that the data and evidence support the alternative hypothesis more than the null hypothesis. When conducting statistical analyses, it is crucial to consider the level of confidence in the results obtained. A commonly used level of confidence is 95%, which means that there is a 5% chance that the conclusion drawn from the statistical analysis is incorrect. This level of confidence also applies when rejecting the null hypothesis. In other words, there is still a 5% chance that, despite rejecting the null hypothesis, the conclusion drawn may not be accurate. Moreover, the alternative hypothesis may not fully explain the phenomenon under study. In some cases, another alternative hypothesis may provide a better explanation or there may be some unknown factors influencing the outcome, making it challenging to definitively prove the alternative hypothesis to be true. Hence, it is inappropriate to state that the alternative hypothesis is proven to be true if the null hypothesis is rejected. Instead, it is more accurate to state that there is enough evidence to support the alternative hypothesis. Question 4 If the probability of a type II error in a hypothesis test is 0.10, what is the power of the hypothesis test? Assuming your sample is unbiased, would increasing the size of the sample increase or decrease the power of the hypothesis test? Answer The power of a hypothesis test is the probability of correctly rejecting a null hypothesis when it is false. In other words, it is the ability of a test to detect an effect or difference if it truly exists. A type II error, on the other hand, is when a false null hypothesis is not rejected. Hence, the power of a hypothesis test is directly related to the probability of a type II error. In this case, if the probability of a type II error is 0.10, then the power of the hypothesis test would be 1 - 0.10 = 0.90, or 90%. This means that the test has a 90% chance of correctly detecting a significant effect or difference if it exists. This can also be written as P(Type II error) = 0.10, P(Type II error) = 1 - Power. Assuming that the sample is unbiased, increasing the size of the sample would increase the power of the hypothesis test. This is because a larger sample size reduces the variability of the data, making it easier to detect a significant effect or difference. In other words, increasing the sample size decreases the probability of a type II error, which in turn increases the power of the test. One way to calculate the power of a hypothesis test is to use the power function, which is given by: power = 1 - P(Type II error) = 1 - Beta = 1 - P(Z < Z_critical + (alpha/SD)) Where Z_critical is the critical value of the test statistic, alpha is the level of significance, and SD is the standard deviation of the population. Another approach is to use power analysis, which involves determining the minimum sample size needed to achieve a desired power level for a given effect size and level of significance. This can be done using statistical software or online calculators. Question 5 You wish to test a directional, one-sided null hypothesis using an independent sample ttest. You run the t-test in SPSS (or EXCEL), and the output states “Sig. 2-tailed = 0.074.” If your hypothesis test was testing at a 5% (.05) significance level, what would your response be regarding the null hypothesis? How would you formally state your results? Answer Based on the provided information, it can be inferred that the p-value obtained from the ttest is 0.074. This means that the probability of obtaining the observed results (or more extreme results) under the assumption of the null hypothesis being true is 0.074. In other words, the observed results are not statistically significant at the 5% level of significance, which was set for the hypothesis test. In this case, the null hypothesis cannot be rejected as the obtained p-value (0.074) is greater than the pre-determined level of significance (0.05). This implies that there is not enough evidence to conclude that the sample mean is significantly different from the hypothesized population mean. Formally, the results can be stated as “The null hypothesis cannot be rejected as the pvalue obtained from the t-test (0.074) is greater than the pre-determined level of significance (0.05).” This statement should be accompanied by the direction of the null hypothesis and the specific test statistic and degrees of freedom used in the t-test.