Statistics Answer Note #10

Question 1 how do you determine there were 9 positive cases when the text mentioned onlt culure of anesthesist E was positive Answer To determine the number of positive cases, several factors must be considered. Firstly, it is important to understand the type of test used to detect the presence of the microorganism in question. In this case, it is not clearly stated if it was a qualitative or quantitative test. If it was a qualitative test, it would only indicate the presence or absence of the microorganism and cannot determine the number of positive cases. However, if it was a quantitative test, it would provide a numerical value that can be used to determine the number of positive cases. In addition, it is also necessary to know the sample size and the percentage of positive results. Without this information, it is not possible to accurately determine the number of positive cases. For example, if the culture of anesthesiologist E was the only positive out of a total of 10 cultures, then it can be assumed that there were 9 negative cases. On the other hand, if the culture of anesthesiologist E was positive out of a larger sample size, such as 100 cultures, then there may have been more positive cases among the remaining cultures. Furthermore, it is important to consider the possibility of false positives and false negatives in the test results. False positives occur when the test indicates a positive result, but the individual does not actually have the microorganism. False negatives, on the other hand, occur when the test indicates a negative result, but the individual does have the microorganism. It is crucial to account for these possibilities in order to accurately determine the number of positive cases. One approach to determining the number of positive cases is to use the formula: number of positive cases = (total number of cases * percentage of positive results) - number of false positives. This formula takes into account the sample size and percentage of positive results, as well as the potential for false positives. Question 2 (c) Find the p-value for F = 4.61, df(factor) = 2, df(error) = 14. give the upper and lower answer bounds exactly Answer Based on the given information, we have an F-statistic of 4.61 with degrees of freedom for the factor (k) being 2, and degrees of freedom for error (n-k) being 14. This question pertains to a one-way ANOVA test, where we are testing for differences between means of three or more groups. To find the p-value, we first need to determine the critical value of F from a table. The critical value for F at a significance level of 0.05 and degrees of freedom 2 and 14 is 3.59. Since our F-statistic of 4.61 is higher than the critical value, we can reject the null hypothesis and conclude that there are significant differences between the means of the groups. The p-value for F can be calculated by finding the area under the curve to the right of the F-statistic on an F-distribution curve with degrees of freedom 2 and 14. This can be easily done using statistical software or by using the Excel function "FDIST". When we plug in the values of F=4.61, df=2, and df=14, we get a p-value of 0.026. Answer bounds: Upper bound: p-value = 0.026 Lower bound: p-value = 0.00 Question 3 We are 95% confident the proportion of those in favor of increasing taxes is between (0.163, 0.577). What is the point estimate of the population proportion? Answer Based on the given information, it can be inferred that a sample was taken and the confidence interval of 95% was calculated based on that sample to estimate the true proportion of the entire population in favor of increasing taxes. In other words, this confidence interval is a range of values within which the true population proportion is expected to lie with a 95% probability. To determine the point estimate of the population proportion, we need to find the midpoint or the average of the lower and upper bounds of the confidence interval. This can be calculated by adding the lower and upper bounds and dividing the sum by 2. Point estimate = (0.163 + 0.577)/2 = 0.37 Therefore, the point estimate of the population proportion in favor of increasing taxes is 0.37 or 37%. Question 4 Construct a 95% confidence interval of the population mean when top enclose x = 25, sigma equals 8, and n = 4. We are 95% confident the population mean is between what and what? (Round to four decimal places.) Answer According to the Central Limit Theorem, for a sample size of n = 4, if the population standard deviation (sigma) is known and the sample mean is normally distributed, we can use the following formula to construct a 95% confidence interval for the population mean (mu): CI = X̄ ± Zα/2 * (σ/√n) Where X̄ is the sample mean, Zα/2 is the critical value for a 95% confidence interval (which is 1.96), σ is the population standard deviation and n is the sample size. Substituting the given values, we get: CI = 25 ± 1.96 * (8/√4) = 25 ± 3.92 Therefore, the 95% confidence interval for the population mean with these given values is between 21.08 and 28.92. To further understand this concept, here are some useful sources that cover the construction of confidence intervals and their interpretations: 1. "Confidence Intervals" - Khan Academy: This source provides a comprehensive explanation of confidence intervals in statistics, including the formula and examples of different types of confidence intervals. It also covers the interpretation of confidence intervals. 2. "Confidence Intervals for a Sample Mean" - University of Sheffield: This source delves into the calculation of confidence intervals for a sample mean, providing step-by- step instructions along with an in-depth explanation of the concept. 3. "Introduction to Confidence Intervals" - Iowa State University: This source offers a detailed overview of confidence intervals, including their definition, properties, and construction. It also provides examples and exercises for practice. In conclusion, using the given values of top enclosure (X) = 25, population standard deviation (sigma) = 8, and sample size (n) = 4, we can construct a 95% confidence interval for the population mean to be between 21.08 and 28.92. This interval gives us a high level of confidence that the population mean lies within this range. Question 5 According to an agricultural study, 80% of apple trees in a particular orchard produce fruit yearly. Given these conditions, what is the probability that exactly 100 of the 120 apple trees in the orchard will produce fruit this year? Answer According to the given information, we can assume that the probability of an apple tree producing fruit in a given year is 0.8 or 80%. This means that out of 120 apple trees, we can expect 96 of them to produce fruit (120 x 0.8 = 96). To calculate the probability of exactly 100 trees producing fruit, we can use the binomial distribution formula: P(x) = (nCx) * p^x * (1-p)^(n-x) Where: P(x) = Probability of exactly x success n = Number of trials p = Probability of success in one trial x = Number of successes In our case: n = 120 p = 0.8 x = 100 P(100) = (120C100) * 0.8^100 * (1-0.8)^(120-100) = 0.00000003 Therefore, the probability of exactly 100 apple trees producing fruit this year is 0.000003 or 0.003%. This is a very small probability, which means that it is highly unlikely for exactly 100 trees to produce fruit in the given orchard. In conclusion, the probability of exactly 100 of the 120 apple trees in the given orchard producing fruit this year is extremely low. This concept can be understood and explored in depth through the study of probability and statistics in the agricultural context.