Statistics and Probability Answers

In R studio
# 1. Monty-Hall Three doors

Recall the Monty-Hall game with three doors, discussed in class. Run a simulation to check that the probablility of winning increases to 2/3 if we switch doors at step two.

Set up the experiment two functions "monty_3doors_noswitch" and "monty_3doors_switch" (these functions will have no input values):

```{r}
monty_3doors_noswitch <- function(){

}

monty_3doors_switch <- function(){

}

```

Use your two functions and the replicate function to compute the empirical probablility of winning for the two experiments.
Compare your answers with the actual theoretical predictions.

```{r}

```

Let X~Bin(50, 0.64). For samples of size 36, calculate P(31.28 < X bar < 33.36).

5. Suppose that a random sample of 25 mining caps was tested from a population of 300 mining caps and 20 exploded properly. Construct a 95% confidence interval for the proportion of mining caps that will explode properly.

1. It is known that the average price for a cappuccino in a restaurant is R15.40 with a standard deviation of R1.20. What is the probability that the sampling error for the mean is less than 0.20 for a sample of 32 restaurants?

4. If a random sample of 50 cups of soda dispensed from a machine showed the mean amount of soda dispensed was 7.9 ounces with s=0.35 ounces, find the confidence level if the margin of error is 0.083.
5. Suppose that a random sample of 25 mining caps was tested from a population of 300 mining caps and 20 exploded properly.

1. It is known that the average price for a cappuccino in a restaurant is R15.40 with a standard deviation of R1.20. What is the probability that the sampling error for the mean is less than 0.20 for a sample of 32 restaurants?

2. It is known that, when given a choice between ordering an espresso or a cappuccino, 63% of people will order the cappuccino. For a sample of 253 people, what is the probability that sampling error for the proportion will be greater than 0.05?

3. Fifteen employees of a large manufacturing company were involved in a study to test a new production method. The mean production rate for the sample of 15 employees was 63 components per hour and the standard deviation was 8 components per hour. Calculate the margin of error for the mean number of components produced per hour at a 5% level of significance.

1. Let X~Bin(50, 0.64). For samples of size 36, calculate P(31.28 < X bar < 33.36).

The Food Marketing Institute shows that 17% of households spend more than $100 per week on groceries. If we assume that this is the population proportion, and a simple random sample of size 800 is taken, then calculate the following.

2. The expected value of p bar, the sample proportion of households that spend more than $100 per week on groceries.

3. The standard deviation of p bar, the sample proportion of households that spend more than $100 per week on groceries.

4. What is the probability that the sample proportion will be within 0.02 units of the population proportion?

5. Repeat question 4 if a sample of 1200 households is taken.

. Suppose that the time in days until hospital discharge for a certain patient population follows a density f(x) = (1/3.3)exp(−x/3.3) for x > 0.
a. Find the mean and variance of this distribution.
b. The general form of this density (the exponential density) is f(x) =(1/β)exp(−x/β) for x > 0 for a fixed value of β. Calculate the mean and variance of this density.

2.. A BTC operator on the average handles 20 calls every 10 minutes. It is known that
the number of calls received in a time interval follows Poisson distribution. Find the
probability that the operator will
a) be busy in the next one minute
b) handle at least two calls in the next two minutes
c) handle at least one call in the next 90 seconds.