A random sample of 20 widgets was examined, 4 widgets out of these 20 are found to be defective. What is the probability that the machine had been shaken? Hint: What is the probability of finding 4 defective widgets in a sample of 20, if (given) the machine has been shaken? What is the probability of finding 4 defective widgets in a sample of 20, if (given) the machine has not been shaken? (b)

Solve In R programmning language: Calculate the probability for each of the following events: (a) A standard normally distributed variable is less than -2.5. (b) A normally distributed variable with mean 35 and standard deviation 6 is larger than 42 but less than 45. (c) A normally distributed variable with mean 35 and standard deviation 6 is larger than 40 but less than 41. (d) X < 0.9 when X has the standard uniform distribution (min=0, max=1). (e) 1 < X < 3 in the exp distribution with rate λ = 2.

A detective has interviewed four witnesses to a crime. From the stories of the witnesses the detective has concluded that:     If the handyman is lying, then so is the butler.   The gardener and the handyman cannot both be telling the truth.     The cook and the gardener are not both lying.     The handyman is telling the truth only if the cook is lying.  For each of the four witnesses, can the detective determine whether that person is telling the truth or lying? Use the truth table to explain your reasoning (Give the full truth table). Please use the following notations in your reasoning –     B: The butler is telling the truth.     C: The cooking is telling the truth.     G: The gardener is telling the truth.     H: The handyman is telling the truth.

consider a population where 30% is in poverty. If you randomly sampled 10 people, how many would you expect to be in poverty?

Explain how we perform maximum likelihood

Step 4: Hypothesis Test for the Population Mean (II) A team averaging 110 points is likely to do very well during the regular season. The coach of your team has hypothesized that your team scored at an average of less than 110 points in the years 2013-2015. Test this claim at a 1% level of significance. For this test, assume that the population standard deviation for relative skill level is unknown. You are to write this code block yourself. Use Step 3 to help you write this code block. Here is some information that will help you write this code block. Reach out to your instructor if you need help. 1. The dataframe for your team is called your_team_df. 2. The variable 'pts' represents the points scored by your team. 3. Calculate and print the mean points scored by your team during the years you picked. 4. Identify the mean score under the null hypothesis. You only have to identify this value and do not have to print it. (Hint: this is given in the problem statement) Assuming the…

Suppose you have to select one project partner from a set of four classmates, who have different GPAs. Assume you do not know any student’s GPA in advance but can get to know it after you have picked a student from that group (a) Suppose you pick one of the four students at random and accept that student as your project partner. What is the probability that your partner is the one with the highest GPA? (b) Suppose you decide to reject the first student and to then accept the next student if and only if that student has a higher GPA. Note that you MUST have a partner, so if the first three are rejected by you, then you have to accept the fourth student. What is the probability that your partner will be the one with the highest GPA.