Calculate the following Manually and Use R codes to prove if necessary.
1. The caffeine content (in milligrams) of a random sample of 50 cups of black coffee dispensed by a new machine is measured. The mean from the sample is 96 mg with a variance of 50.41mg2 The machine is capable of dispensing 3000 cups per day, whose caffeine content is normally distributed. Test at 5% significance level whether the mean caffeine content in a cup of black coffee dispensed by this machine is less than 100mg. a) The p-value of the test is? b) The Critical value of the test is ? c) The calculated test statistic is ? 2. A single leaf was taken from each of 11 different tobacco plants. Each was divided in half; one half was chosen at random and treated with preparation | and the other half with preparation II. The object of the experiment was to compare the effects of the two preparations of mosaic virus on the number of lesions on the half leaves after a fixed period of time. The data are recorded in the table below. Test at 5% significance level whether the distributions of lesions are different. A) The p-value of the test is? B) The absolute value of the Critical value of this test is? C) The absolute value of the calculated test statistic is ? Number of lesions on half leaf. Tobacco plant Preparation 1 Preparation 2 1 18 14 2 20 15 3 9 6 4 14 12 5 38 30 6 26 32 7 15 9 8 10 2 9 25 18 10 7 3 11 13 6 3. A sample of 15 people participate in a study which compares the effectiveness of two drugs for reducing the level of the LDL (low density lipoprotein) blood cholesterol. After using the first drug for two weeks, the decrease in their cholesterol level is recorded as the G measurement. After a pause of two months, the same individuals are administered another drug for two weeks, and the new decrease in their cholesterol level is recorded as the H measurement. The Table below gives the measurements in mg/dl. Test at 20% significance level whether one of the drugs is more effective than the other. A) The p-value of the test is ? B) The absolute value of the Critical value is? C) The absolute value of the calculated test statistic is ? G 13.1 12.3 10.0 17.7 19.4 10.1 11.5 12.0 9.5 12.1 18.0 7.5 6.9 14.5 8.6 H 12.0 7.3 11.7 12.5 18.6 12.3 15.2 16.1 10.7 9.8 15.3 6.4 8.5 16.4 7.8 4. The pH level of the soil between 5.3 and 6.5 is optimal for strawberries. To measure the pH level, a field is divided into two lots. In each lot, we randomly select 20 samples of soil. The data are given below. Assume that the pH levels of the two lots are normally distributed. Test at 10% significance level whether the two slots have different variances. A)The p-value of the test is? B) The calculated test statistic is ? Lot 5.66 5.73 5.68 5.77 5.73 5.71 5.98 5.58 6.11 5.37 5.76 5.59 5.94 | 5.84 5.53 5.64 1 5.73 5.30 5.65 5.70 Lot 5.62 6.03 6.05 5.91 5.63 6.10 5.89 6.06 5.73 5.64 5.84 6.16 5.64 6.00 5.79 5.91 6.17 6.02 5.76 6.07 ii) Assuming the two variances are equal , test at the 0.5% significance level whether the 2 lots have different average PH. A) The p-value of the test is ? B) The absolute value of the Critical value is ? The absolute value of the calculated test statistic is
