## Probability and Inferential Statistics Project Worksheet.

Probability and Inferential Statistics Project Worksheet.

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MAT 274 BENCHMARK FORMAT AND STYLE TEMPLATE

Since part 1 and 2 depends on your ID number, you need to clearly state the last digit of your GCU student ID number at the beginning of your project.

- A patient is classified as having gestational diabetes if their average glucose level is above 140 milligrams per deciliter (mg/dl) one hour after a sugary drink is ingested. Rebecca’s doctor is concerned that she may suffer from gestational diabetes. There is variation both in the actual glucose level and in the blood test that measures the level. Rebecca’s measured glucose level one hour after ingesting the sugary drink varies according to the Normal distribution with μ=140+# mg/dl and σ=#+1 mg/dl, where # is the last digit of your GCU student ID number. Using the Central Limit Theorem, determine the probability of Rebecca being diagnosed with gestational diabetes if her glucose level is measured:
- Once?
- n=#+2 times, where # is the last digit of your student ID?
- n=#+4 times, where # is the last digit of your student ID?
- Comment on the relationship between the probabilities observed in (a), (b), and (c). Explain, using concepts from lecture why this occurs and what it means in context.

Part a, b, c requires a calculation using normal distribution.

You do not have to provide the sketch.

However, you need to provide the screen shot of Excel computation part in addition to the written answer.

To practice a similar problem, Check Topic 3 homework No 52( 6.4.6-T). It is a similar question.

To use Excel Template, use the NormalCalculator tab, which can be found in the very first sheet. Then, just copy and paste the portion like below. Of course, you need to write the answer in your own writing to answer the question. Please remember that you need to take the sample size into consideration when you type standard deviation. In part d, you need to write some comments following the instruction.

Given Data Values and Mean/Standard Deviation | |

Data Value (x) | 1.1200 |

Standard Deviation | 1.0000 |

Mean | 0.0000 |

Z-Score | 1.1200 |

Left Probability | 0.8686 |

Right Probability | 0.1314 |

For each part, insert your sketch of the required area under the normal curve. In addition, include a screenshot of your Excel computation to find this area.

- Insert screenshot and figure for part (a)

- Insert screenshot and figure for part (b)

- Insert screenshot and figure for part (c)

- Comment on the relationship among the probabilities in parts (a),(b), and (c).

- Suppose next that we have even less knowledge of our patient, and we are only given the accuracy of the blood test and prevalence of the disease in our population. We are told that the blood test is 9# percent reliable, this means that the test will yield an accurate positive result in 9#% of the cases where the disease is actually present. Gestational diabetes affects #+1 percent of the population in our patient’s age group, and that our test has a false positive rate of #+4 percent. Use your knowledge of Bayes’ Theorem and Conditional Probabilities to compute the following quantities based on the information given only in part 2:
- If 100,000 people take the blood test, how many people would you expect to test positive and actually have gestational diabetes?
- What is the probability of having the disease given that you test positive?
- If 100,000 people take the blood test, how many people would you expect to test negative despite actually having gestational diabetes?
- What is the probability of having the disease given that you tested negative?
- Comment on what you observe in the above computations. How does the prevalence of the disease affect whether the test can be trusted?

Fill in the conditional probability table here, then answer the questions in each part below. You need to fill out the table using the following format. For similar exercise problems, check Topic 2 homework No 38(4.3.13), 39(4.3.18), 40(4.3.23). Please remember that part a and c answers come directly from the table after filling out. Part b and d requires the conditional probability. You still need to use the numbers from the table.

When you fill out the table, first start with Total for Disease using diabetes affects #+1 percent of the population in our patient’s age group. Then, you need to fill out the rest of the table.

In part v, you need to discuss following the given instruction.

Positive | Negative | Total | |

Disease | |||

No Disease | |||

Total | 100,000 |

- Answer part (a) here.

- Answer part (b) here.

- Answer part (c) here.

- Answer part (d) here.

- Comment on how prevalence of the disease affects your ability to trust the test. Discuss what factors would lead you to trust the blood test, or not trust the blood test.

- As we have seen in class, hypothesis testing, and confidence intervals are the most common inferential tools used in statistics. Imagine that you have been tasked with designing an experiment to determine reliably if a patient should be diagnosed with diabetes based on their blood test results. Create a short outline of your experiment, including all the following:

In part a, b, c, d, you are expected to design your experiment and provide hypotheses.

Provide as much details as possible. This is kind of research proposal.

If you have enough time, you may actually carry out your research and provide results in part e and f. However, it might not be feasible to do that in a limited time.

Hence, you may just talk about those items in part e and f. For example, you may not have actual lower limit and upper limit, but can discuss what those items mean in the context of your research.

You need to discuss all items in part e and f. For the last item(make a decision on the null.), you can talk about what we can do about the null hypothesis based on p-value. You may not have an actual p-value, but you can talk about the decision on the null hypothesis, for example, if p-value is less than or equal to …, then …… If p-value is greater than …., then …..

For null hypothesis, you can just use H0. For alternative hypothesis, you can use H1.

- A detailed discussion of your experimental design. Detailed experimental design should include the type of experiment, how you chose your sample size, what data is being collected, and how you would collect that data.

- How is randomization used in your sampling or assignment strategy? Remember to discuss how you would randomize for sampling and assignment, what type of randomization are you using?

- The type of inferential test utilized in your experiment. Include type of test used, number of tails, and a justification for this choice.

- A formal statement of the null and alternative hypothesis for your test. Make sure to include correct statistical notation for the formal null and alternative, do not just state this in words.

- A confidence interval for estimating the parameter in your test. State and discuss your chosen confidence level, why this is appropriate, and interpret the lower and upper limits.

- An interpretation of your p-value and confidence interval, including what they mean in the context of your experimental design. Answer each part below. State your significance level, interpret your p-value, and make a decision on the null.