Running Head: Hypothesis Testing. 1 Hypothesis Testing Mary Gibson 01-23-2019 Hypothesis Testing. 2 Steps of Hypothesis Testing. 1. State the null and alternative hypothesis (H0 and H1) 2. State the level of significance. For example, ∝=0.05. The purpose of hypothesis testing is not to question the computed value of the sample statistics but to make a judgement about the difference between the sample statistic and a hypothesized sample parameter. Therefore, we need to make a decision on the criterion to use for deciding whether to accept or reject the null hypothesis. A higher significance level means the chances of dropping the null hypothesis when it is true are high. Decide which distribution to use in hypothesis testing. That is the test statistic (Efron, 2014). 3. Determine the critical region or regions. 4. Compute the test statistic value. 5. Make a statistical decision. That is, decide on whether to accept or reject the null hypothesis depending on whether test statistic lies on then critical regions. 6. Finally, make the management decision. When performing hypothesis testing the best method to use is the critical value method. The critical value method or the critical region method is test statistic that lead to the rejection of the null hypothesis form the critical region of a test while those that lead to acceptance of null hypothesis form the acceptance region (Johansen, 2011). Consider the diagram below to clearly understand the critical value method. Hypothesis Testing. If the ∝ is 0.05 the rejection region will lie in the shaded region and the acceptance region will lie in the unshaded region. There are two critical regions whose area sum is equal to the level of significance∝. Performing hypothesis testing. α = 0.05 Claim; age of the patients is less than 65 years. Step 1; Statement of the hypothesis. H0 = 65 H1 < 65 Step 2; This is a right tailed test with α = 0.05. Consider the diagram below to support the argument and facilitate understanding. 3 Hypothesis Testing. 4 This is a right tailed test because the rejection region lies in the shaded region meaning that the claim is less than 65. Any value which lies in the rejection region means that we will reject the null hypothesis (Johansen, 2011). Step 3; We are going to use Z-test statistic because this is a large sample. The sample size is greater than 30 (that is, n=60), thus it’s a large sample and use Z-test statistic. T-test statistic uses a small sample size, which is less than 30 (n< 30) Step 4; Table area = 0.05-0.01=0.49 From the standard normal tables the critical value, Z α = Z0.01 = 2.33 Step 5; Test statistic. Since the population standard deviation, ∂ is unknown we use sample standard deviation, S S=8.779, Ẋ=61.917, µ=65, n=60 Hypothesis Testing. The test statistic is Z= = 5 Ẋ−µ 𝑆 √𝑛 = Ẋ−µ 𝜕 √𝑛 61.917−65 8.779 √60 = -2.72 Step 6; statistical decision. Since the sample test statistic Z = -2.72 does not lie in the critical region we accept the null hypothesis. Step 7; interpretation. Therefore the average age of the patients who visit the hospital is not less than 65. Conclusion. From week one we carried out a number of statistical computations in search of information that will help us make a decision for the hospital patients. From week one we saw that the average age of the patients who visit the hospital is 61.917 years. The modal age came at 70.75 years. This was not enough statistical information to draw a conclusion for the management of the hospital. We had to further perform some statistical hypothesis to test for some claims made by the management. This hypothesis testing was based on a specific level of significance. The hypothesis testing revealed that the average of the patients who visit NCLEX Memorial Hospital is not less than 65. Therefore the management should employ a policy that will put more emphasis on treating patients who have 65 years and above, since this is the age bracket that has more patients infected with infectious disease (Efron, 2014). Hypothesis Testing. 6 References Efron, B. (2014). Large-scale simultaneous hypothesis testing: the choice of a null hypothesis. Journal of the American Statistical Association, 99(465), 96-104. Johansen, S. (2011). Estimation and hypothesis testing of co-integration vectors in Gaussian vector autoregressive models. Econometric: Journal of the Econometric Society, 15511580.

Patient # Infectious Disease Age 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes 69 35 60 55 49 60 72 70 70 73 68 72 74 69 46 48 70 55 49 60 72 70 76 56 59 64 71 69 55 61 70 55 45 69 54 48 60 61 50 59 60 62 63 53 64 Descriptives Mean Median Mode Midrange Range Variance Standard deviation 95% Confidence interval - mean 99% Confidence interval - mean 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes 50 69 52 68 70 69 59 58 69 65 61 59 71 71 68 Age 61.82 61.5 69 55.5 41 79.64 8.92 fidence interval - mean 95% 61.82 8.92 60 1.960 2.2581 64.0748 59.5585 confidence level mean std. dev. n z half-width upper confidence limit lower confidence limit fidence interval - mean 99% 61.82 8.92 60 2.576 2.9677 64.7843 58.8490 confidence level mean std. dev. n z half-width upper confidence limit lower confidence limit

Running head: CONFIDENCE INTERVAL 1 Confidence interval Mary Gibson CONFIDENCE INTERVAL 2 Confidence interval Confidence interval expresses a type of interval estimate in statistics that gives a range of values which have a likelihood of containing a population parameter. The interval has an associated degree of confidence, which implies a certain likelihood that a population parameter is within the confidence interval. Thus, the confidence interval gives the range of values that are likely to contain the true parameter of the population (Bland, 2015). Point estimate A point estimate expresses a single value that is evaluated from sample data to represent the best guess for an unknown parameter of a population. Notably, the sample mean is best point estimate for the true mean of the population since its gives the average of a data sample obtained from the given population. Confidence intervals are important in decision making since they reduce the risk of making an error by restricting the real answer with a certain estimated range. Point estimate for the population mean of the dataset Using the Option #2 dataset, the point estimate is the sample mean of the age of the patients which is 61.82 years. 99% and 95% confidence intervals The confidence intervals can be evaluated by the formula (Bland, 2015); 𝐶𝑜𝑛𝑓𝑖𝑑𝑒𝑛𝑐𝑒 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 = 𝑥̌ ± 𝑧 ∗ 𝑠 √𝑛 CONFIDENCE INTERVAL 3 Where 𝑥̌ represents the sample mean, z represents the z-score of the degree of confidence, s represents the standard deviation of the sample, and n represents the sample size. Using the excel; the 95% confidence interval can be given as; 95% Confidence interval – mean 95% 61.82 8.92 60 1.960 2.2581 64.0748 59.5585 confidence level Mean std. dev. N Z half-width upper confidence limit lower confidence limit Thus, the 95% confidence interval is (59.56, 64.07). This implies that we can be 95% confident that the population mean for the age of patients is between 59.56 years and 64.07 years. The 99% confidence interval can be obtained as; 99% Confidence interval – mean 99% 61.82 8.92 60 2.576 2.9677 64.7843 58.8490 confidence level Mean std. dev. N Z half-width upper confidence limit lower confidence limit CONFIDENCE INTERVAL 4 The 99% confidence interval is (58.85, 64.78). This implies that we can be 99% confident that the population mean for the age of the patients is between 58.85 years and 64.78 years. Interpretation The 95% confidence interval is (59.56, 64.07) whilst the 99% confidence interval is (58.85, 64.78). As such, the 99% confidence interval is wider compared to the 95% confidence interval. This implies that as the degree of confidence increased, the interval estimate widened. The increase in confidence interval implies that the chances of making an error in estimating the population mean decreases since it can be observed from a wider range of values. Increase in confidence level widens the interval estimate thus reducing the risk of making an error in estimation. CONFIDENCE INTERVAL References Bland, M. (2015). An introduction to medical statistics. Oxford University Press (UK). 5

Patient # Infectious Disease Age 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes 69 35 60 55 49 60 72 70 70 73 68 72 74 69 46 48 70 55 49 60 72 70 76 56 59 64 71 69 55 61 70 55 45 69 54 48 60 61 50 59 60 62 63 53 64 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 Mean Median Mode Midrange Range Variance Standard Deviation Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes 61.82 61.50 69.00 58.50 41.00 79.64 8.30 50 69 52 68 70 69 59 58 69 65 61 59 71 71 68 76 35