a. Tom designed a complete factorial experiment with 2 factors. One factor had 4 levels, the other had 5 levels. For each combination of levels of factors, there were 6 replicates. In the ANOVA table associated with this design, what were the degrees of freedom of the interaction?

a. 7, b. 12,   c. 20, d. 24, e. 30.

b. In the above ANOVA table of the complete factorial experiment, suppose we want to test the significance of the main effect of the factor with 4 levels. What is the distribution we use to calculate p-values?              

a. F(3, 100), b. F (4, 120),    c. F (4, 100), d. F (5, 100), e. F (5, 120).

Show all calculations in a neat and organized manner. Be sure that I understand your logic. For your experiment you must treat 1 X 109 cells with 5 nM cycloheximide (MW=281.35). Cycloheximide is both toxic and expensive. You have counted the cells with a hemocytometer and found the following numbers when counting 5 “4x4” sections (each 4x4 is 0.004 µl) of a 1:10 dilution of you chlamy stock: 90, 103, 123, 99, 107.

1) What is the concentration of the initial Chlamy culture in cell/ml? _______________________

2) How many mls of cells of the culture must you use for the experiment? _____________________

3) Describe how you will treat the cells with cyclohexamide. Clearly show all calculations.

USE R-studio TO WRITE THE CODES!

# 2. More Coin Tosses
Experiment: A coin toss has outcomes {H, T}, with P(H) = .6.
We do independent tosses of the coin until we get a head.

Recall that we computed the sample space for this experiment in class, it has infinite number of outcomes.

Define a random variable "tosses_till_heads" that counts the number of tosses until we get a heads.
```{r}

```

Use the replicate function, to run 100000 simulations of this random variable.
```{r}

```
Use these simulations to estimate the probability of getting a head after 15 tosses. Compare this with the theoretical value computed in the lectures.
```{r}

```
Compute the probability of getting a head after 50 tosses. What do you notice?
```{r}

USE R TO WRITE THE CODES!

# 2. More Coin Tosses
Experiment: A coin toss has outcomes {H, T}, with P(H) = .6.
We do independent tosses of the coin until we get a head.

Recall that we computed the sample space for this experiment in class, it has infinite number of outcomes.

Define a random variable "tosses_till_heads" that counts the number of tosses until we get a heads.
```{r}

```

Use the replicate function, to run 100000 simulations of this random variable.
```{r}

```
Use these simulations to estimate the probability of getting a head after 15 tosses. Compare this with the theoretical value computed in the lectures.
```{r}

```
Compute the probability of getting a head after 50 tosses. What do you notice?
```{r}

Sorry for the layman question, but it's not my field.

Suppose this thought experiment is performed. Light takes 8 minutes to go from the surface of the Sun to Earth. Imagine the Sun is suddenly removed. Clearly, for the remaining 8 minutes, we won't see any difference.

However, I am wondering about the gravitational effect of the Sun. If the propagation of the gravitational force travels with the speed of light, for 8 minutes the Earth will continue to follow an orbit around nothing. If however, gravity is due to a distortion of spacetime, this distortion will cease to exist as soon as the mass is removed, thus the Earth will leave through the orbit tangent.

What is the state of the art of research for this thought experiment? I am pretty sure this is knowledge that can be inferred from observation.

Steven says that the period T of an object on a spring is depends on the mass m of the object and the spring constant k of the spring in the following way: T=2π k m ​  . Jessica says that the period T of an object on a spring is depends on the mass m of the object and the spring constant k of the spring in the following way: T=2π k m ​ ​  .

1.What are two different ways you can test and which of the two mathematical models above is correct? Note: the one spring has a spring constant that is twice as big as the other. Once you've done brainstorming, describe your two experiments in detail in the text box below.

Hint: Make sure you address the following points in your discussion: a. What are the mathematical models you're testing? b. What quantities can you vary or change to test the two models? c. What quantities will you measure and how will you measure them?

2.For each of your experiments, describe the 2 predicted outcomes of the experiment based on the 2 mathematical models that you're testing. Include a discussion of any simplifying assumptions about the experiment that you're making in making each prediction. Hint: a. A prediction should be based on the model being tested. One experiment needs 2 predictions because you're testing two competing models. b. An assumption is an experimental factor that you're choosing to ignore in applying the mathematical model to make a prediction. 3.Discuss:

3.What is/are the source(s) of experimental uncertainty in each of your experiments? How will you minimize these uncertainties in your experimental design? For EACH experiment: If you are using measured quantities to make a prediction, estimate the uncertainty in your predicted quantity using the weakest link rule.

In this experiment the concentration of Vitamin C/ascorbic acid is used to find the amount of iodine present through titration and the stoichiometric ratio. Furthermore, by finding the amount of iodine present before and after the equivalence point, the experiment presents us with the amount of iodine that reacts with the Vitamin C in “unknown B”.There were two titrations performed in this experiment, three times each. The first began with 50mL of Sodium Thiosulphate put in a burette. Approximately 2g of Potassium Iodide was stirred in a 250mL Erlenmeyer flask, with 10mL of 0.3M Sulphuric Acid, followed by 20mL of 1.0003x10-2M Potassium Iodate. The solution was initially a brown color, titrated until the solution becomes lighter. 1mL of starch indicator to the solution turned the lighter solution blue, and was titrated until the blue disappeared. The second titration began with 50mL of Sodium Thiosulphate in a burette. Placing approximately 2g of Potassium Iodide in a 250mL Erlenmeyer flask, 20mL of “Pink Drink”, with 10mL of 0.3M Sulphuric Acid, and 20mL of 1.0003x10-2M Potassium Iodate was stirred together. The brown mixture was titrated until it lightened; with 1mL of starch indicator added it changed to blue. The titration was carried out until the blue disappeared.

Using these equations
IO3 + 5I + 6H = 3I2 + 3H2O
I2 + 2S2O3 = 2I + S4O6

How do I go about finding the amount of vitamin C and iodine? I have equations but I'm told they're the wrong ones. Quite frankly I'm lost. This is the data I collected during the experiment.

You have a large barrel full of coins. 5 percent of the coins in the barrel are “type X” and 95 percent are “type Y.” When you flip them, type X coins come up heads 90 percent of the time and tails 10 percent of the time. Type Y coins come up heads 3 percent of the time and tails 97 percent of the time.

Suppose you do a three part experiment: (i) Take a coin at random from the barrel [in this context, choosing a coin “at random” from the barrel means that there is a 5 percent chance you will select a type X coin and a 95 percent chance you will select a type Y coin] , (ii) flip the coin you selected once and observe whether it comes up heads or tails, and (iii) flip the same coin a second time and observe whether it comes up heads or tails.

a) Represent this experiment in a tree, showing all the possible outcomes, along with the probability of each outcome. [For example, one possible outcome is "the coin is type Y, the first flip is tails, and the second flip is heads," which you could abbreviate as YTH.]

b) What is the probability that the two tosses of the coin do not give the same result? (That is, what is the probability that either the first toss is heads and the second toss is tails, or the first toss is tails and the second toss is heads?)

c) If the two tosses of the coin do not give the same result, what is the probability that the coin chosen at random is type Y?

d) Define event A to be the set of all outcomes of this experiment in which the first flip comes up Heads. Define event B to be the set of all outcomes of this experiment in which the second flip comes up Heads. Are A and B independent events? Justify your answer carefully, using an argument based on the definition of statistical independence, not just intuition.

1. 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. 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. We are told that the blood test is 98 percent reliable and gestational diabetes affects 9 percent of the population in our patient’s age group, and that our test has a false positive rate of 12 percent. Create a short outline of your experiment, including all the following:
   1. 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.
2. 2. 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?
3. 3. The type of inferential test utilized in your experiment. Include type of test used, number of tails, and a justification for this choice.
4. 4. 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.
5. 5. 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.
6. 6. 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.