title: “2B03 Assignment 4”
subtitle: “Statistical Inference (Chapters 7, 8, & 9)”
date: ‘Due Thursday November 16 2023’
format: pdf
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> Instructions: *You are to use for generating your assignment output file. You begin with the script downloaded from A2L, and need to pay attention to information provided via introductory material posted to A2L on working with R and . Having added your answers to the script, you then are to generate your output file using the “Render” button in the RStudio IDE and, when complete, upload both your file and your PDF file to the appropriate folder on A2L.*
1. Define the following terms in a sentence (or *short* paragraph) and state a formula if appropriate (this question is worth 5 marks).
a. Type II Error：Occurs in statistical hypothesis testing when the null hypothesis in correctly not rejected， despite the alternative hypotheses being true.
b. Power of a Test : The power of a statistical test is the probability of correctly rejecting a false null hypothesis. It represents the ability of a test to detect a ture effect or difference.
c. Goodness of Fit Test :
d. $P$-value
e. Simple Regression Analysis
2. A coin operated coffee machine is set to pour 8 oz per cup. A random sample of the weights of a number of cups is as follows: 8.40, 8.25, 8.05, 7.84, 7.36, 8.54, 7.56, 7.56, 8.02, 7.39, 8.34, 8.56.
Test the hypothesis that the machine is delivering at the level set by the manufacturer. Use a 0.01 level of significance (this question is worth 2 marks, you *must* show all steps).
3. Two different brands of milk are randomly sampled, and the fat content in each bottle of milk is determined. Twenty-six bottles of Brand A milk yielded an average fat content of $\bar X_1=25$ grams with $s^2_1=4$, and thirty one bottles of Brand B yielded an average fat content of $\bar X_2=25.8$ grams with $s^2_2=7$ (this question is worth 3 marks, you *must* show all steps).
Test the hypothesis that both brands have identical average fat content at the 5% level of significance.
4. To compare two programs for training industrial workers to perform a skilled job, 20 workers are included in an experiment. Of these, 10 are selected at random and trained by method 1; the remaining 10 are trained by method 2. After completion of training, all the workers are subjected to a time-and-motion test that records the speed of performance of a skilled job. The following time, as measured in minutes, is obtained.
| Method | | | | | | | | | | |
|———-|—–|—–|—–|—–|—–|—–|—–|—–|—–|—–|
| Method 1 | 15 | 20 | 11 | 23 | 16 | 21 | 18 | 16 | 27 | 24 |
| Method 2 | 23 | 31 | 13 | 19 | 23 | 17 | 28 | 26 | 25 | 28 |
Test the hypothesis that the mean job time is equal before and after training with method 1 and 2 versus the alternative that it is significantly less after training with method 1 than after training with method 2. Use a significance level of $\alpha=0.05$ (this question is worth 4 marks, you *must* show all steps).
5. A Canadian-wide marketing survey found that only one-fifth of Canadians drink beer on a regular basis. A random sample of 36 residents in North York found nine who were regular beer drinkers. Test whether or not North York has a *greater* then the national proportion of beer drinkers (i.e. test $H_0\colon\pi= 0.2$ versus $\quad H_1\colon\pi > 0.2$ – this question is worth 3 marks, you *must* show all steps).
6. A firm draws a random sample of 18 ball bearings from the day’s output. The sample variance of their diameters is $s^2=0.009$ inches (this question is worth 4 marks, you *must* show all steps).
a. Construct a 95% confidence interval for the population variance.
b. Construct a 90% confidence interval for the population variance.
c. What assumptions underlie the answers in the first two parts of this question.
7. In an agricultural experiment to determine the effects of a particular insecticide, a field was planted with corn. Half the plants were sprayed with the insecticide, and half were unsprayed. Several weeks later, independent random samples of 200 sprayed plants and 200 unsprayed plants were examined. The number of healthy plants in each sample was as follows (this question is worth 4 marks, you *must* show all steps).
| Status | Sprayed | Unsprayed |
|———–|———|———–|
| Healthy | 131 | 111 |
| Unhealthy | 69 | 89 |
If the significance level is set at $\alpha=0.05$, does the evidence indicate that a higher proportion of sprayed than of unsprayed plants were healthy? Use a one tailed $Z$ test for equality of population proportions (note – since the null is that the proportions are equal, use this information to construct a pooled estimate of the proportion).
8. The success of a federally funded, locally administered manpower program was measured by the proportion of clients who moved from subsidized employment into unsubsidized (private sector) employment and remained there for a certain length of time. A random sample of $n=376$ clients of the program produced the following results (this question is worth 4 marks, you *must* show all steps).
| Education | Success | Failure |
|——————|———|———|
| 8 years or less | 13 | 19 |
| 9 to 11 years | 76 | 45 |
| 12 years | 107 | 65 |
| 13 years or more | 32 | 19 |
a. Estimate the marginal probabilities of success and failure.
b. Test the hypothesis that the program outcomes are independent of educational level using a $\chi^2$ test of independence.
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