3. Let X, Y, and Z be independent unit exponential random variables, with common density f(t) = e^(-t) for t > 0.

Let T_1 = min (X, Y, Z )

T_2 = middle value of the three numbers X, Y, Z

T_3 = max (X, Y, Z )

(a) Find P( T_1 > t ) for t >0.

(b) Find P( T_3 < t ) for t > 0.

(c) Find P( T_2 >  t ) for t > 0.

HINT: T_2 > t happens when how many of X and Y and Z are greater than t ?

(d)  Find E ( T_3 - T_2 ) = expected difference between T_3 and T_2 .

HINT: One way to do part (d) is obviously to find the densities of T_2 and T_3 from the answers to parts (b) and (c) and then to use those densities to calculate E( T_2 ) and E(T_3 ). You could also integrate the survival functions (See page 332, under "Expectation from the survival function"). A much easier way is to just write down the answer, which you can do if you use the memoryless property of exponential distributions. Think about 3 light bulbs with independent unit exponential lifetimes. As long as such a bulb is working, its future behavior is exactly the same as the future behavior of a new bulb.

(e) Find E(T_3) and var(T_3).

HINT: There is almost no work involved in doing part (e) if you figured out the clever way to do part (d) and you use the equality

T_3 = T_1 + (T_2 - T_1) + (T_3 - T_2 ).

. Let X be the soil strength of a sample taken at 35mm depth in a large field. Assume X is normally distributed with an UNKNOWN standard deviation. We are interested in computing confidence intervals for μ, the unknown population mean soil strength at 35mm in this particular field. Four soil samples are randomly chosen from the field with the measurements (15, 21, 30, 34).

       a.     compute the 80% CI.

       b.     compute the 95% CI.

       c.     compare the two confidence intervals from a. and b.

(Which one is more precise? Which one is more accurate

Gun Murders - Texas vs California - Significance Test: California had stricter gun laws than Texas. However, California had a greater proportion of gun murders than Texas. Here we test whether or not the proportion was significantly greater in California. A significant difference is one that is unlikely to be a result of random variation.

The table summarizes the data for each state. The p̂'s are actually population proportions but you should treat them as sample proportions.

The standard error (SE) is given to save calculation time if you are not using software. Data Summary number of total number Proportion State gun murders (x) of murders (n) p̂

California 1220 1786 0.68309

Texas 699 1084 0.64483

SE = 0.01812

The Test: Test the claim that the proportion of gun murders was significantly greater in California than Texas in 2011. Use a 0.05 significance level.

(a) Letting p̂1 be the proportion of gun murders in California and p̂2 be the proportion from Texas, calculate the test statistic using software or the formula z = (p̂1 − p̂2) − δp SE where δp is the hypothesized difference in proportions from the null hypothesis and the standard error (SE) is given with the data. Round your answer to 2 decimal places. z = To account for hand calculations -vs- software, your answer must be within 0.01 of the true answer.

(b) Use software or the z-table to get the P-value of the test statistic. Round to 4 decimal places. P-value =

(c) What is the conclusion regarding the null hypothesis? reject H0 fail to reject H0

(d) Choose the appropriate concluding statement.

The data supports the claim that the proportion of gun murders was significantly greater in California than Texas.

While the proportion of gun murders in California was greater than Texas, the difference was not great enough to be considered significant.

We have proven that the stricter gun laws in California actually increased the proportion of gun murders

A pharmaceutical company has a new drug which relieves headaches. However, there is some indication that the drug may have the side effect of increasing blood pressure. Suppose the drug company conducts a hypothesis test to determine whether the medication raises blood pressure. The hypotheses are:

H₀: The drug does not increase blood pressure.
H_a: The drug increases blood pressure.

Answer the following questions completely:

1. Do you think that for doctors and patients it is more important to have a small α probability or a small β probability? Why?

2. Do you think that the pharmaceutical company would prefer to have a small α probability or a small β probability? Why?

Question 16 options:

1a. Suppose x has a distribution with μ = 21 and σ = 16.

(a) If a random sample of size n = 39 is drawn, find μ_x, σ_x and P(21 ≤ x ≤ 23). (Round σ_x to two decimal places and the probability to four decimal places.)

(b) If a random sample of size n = 55 is drawn, find μ_x, σ_x and P(21 ≤ x ≤ 23). (Round σ_x to two decimal places and the probability to four decimal places.)

A politician claims that medical insurance companies do not cover a majority of the cost and the average patient has to pay more than $10,000 in hospital bills. Using the critical value method, test this claim at the 5% level of significance. Why might having high hospital charges be an issue for patients?

Mean= 13266.97

Standard Deviation= 12110.01

The data from Exhibit 3 is also in the Excel file income.xls on the course website. Use Excel, along with this file, to determine Mrs. Bella’s real income for the last fifteen years. Do this by first converting each price index from percent by dividing by 100. Then, divide gross income by your converted (adjusted) price index. Using Excel, find the mean, median, standard deviation, and variance of her past real income. Explain the meaning of these statistics. Can you use mean income to forecast future earnings? Take into account both statistical and non-statistical considerations.

1. Each month, the owner of Fay's Tanning Salon records in a data file the monthly total sales receipts and the amount spend that month on advertising. (a) Identify the two variables. (b) For each variable, indicate whether it is quantitative or categorical. (c) Identify the response variable and the explanatory variable.

At a Bloomburg City Council meeting, a plan to fund more swim safety programs was presented. The reasoning behind the request was that less than 40% of children under the age of 5 could pass a swim test. If this is true, the council will agree to fund more programs for these kids. The council decides to take a 200-person volunteer sample of children under 5 years in Bloomburg City and conduct a significance test for H0: p = 0.40 and Ha: p < 0.40, where p is the proportion of these children that can pass a swim test. They will perform a significance test at a significance level of α = 0.05 for the hypotheses.

Part A: Describe a Type II error that could occur. What impact could this error have on the situation?

Part B: Out of the 200 children under 5 that volunteered to take a swimming test, 87 passed, resulting in a p-value of 0.8438. What can you conclude from this p-value given the data of the 200 children is sufficient to perform a significance test for the hypotheses?

Part C: What possible defect in the study can you find in Part B? Explain.

The mean and standard deviation for the diameter of a certain type of steel rod are mu = 0.503 cm and sigma = 0.03cm. Let X denote the average of the diameters of a batch of 100 such steel rods. The batch passes inspection if Xbar falls between 0.495 and 0.505cm.

1. What is the approximate distribution of Xbar? Specify the mean and the variance and cite the appropriate theorem to justify your answer.

2. What is the approximate probability the batch will pass inspection?

3. Over the next six months 40 batches of 100 will be delivered. Let Y denote the number of batches that will pass inspection.

(a) the distribution of Y is: Binomial, hypergeometric, negative binomial, OR poisson?

(b) give the approximation, as accurately as possible, to the probability P(Y ≤ 30).

Use the Voltage data to test the claim that home voltages and generator voltages are from populations with the same mean.

1. Create a histogram on the difference in voltage between home voltage and generator voltage. Copy and paste your histogram here.

2. Does our assumption of normality appear to be satisfied?
   1. Yes
   2. No

3. What is your null and alternative?
   1. H0: µ1 = µ2, H1: µ1 ≠ µ2
   2. H0: µ1 ≠ µ2, H1: µ1 = µ2
   3. H0: µ1 = µ2, H1: µ1 > µ2
   4. H0: µ1 > µ2, H1: µ1 = µ2
   5. None of the abov Provide your own answer.

4. What is your decision?
   1. Support H1
   2. Cannot support H1
   3. Cannot make a decision at this time
   4. We do not have enough data to make an educated guess
   5. None of the abov Provide your own answer.

5. What is your conclusion?
   1. We have sufficient evidence to show that the mean voltage is the same for home voltage and generator voltage.
   2. We do not have sufficient evidence to show that the mean voltage is the same for home voltage and generator voltage.
   3. We have sufficient evidence to show the mean voltage is the higher for home voltage than generator voltage.
   4. We do not have sufficient evidence to show that the mean voltage is the higher for home voltage than generator voltage.
   5. None of the abov Provide your own answer.

6. Does the confidence interval support your conclusion?
   1. Yes
   2. No
   3. Cannot determine at this time
   4. None of the above. Provide your own answer.

7. If there is a statically significant difference, does that difference have practical significance?
   1. Yes
   2. No

Home   Generator
123.8   124.8
123.9   124.3
123.9   125.2
123.3   124.5
123.4   125.1
123.3   124.8
123.3   125.1
123.6   125.0
123.5   124.8
123.5   124.7
123.5   124.5
123.7   125.2
123.6   124.4
123.7   124.7
123.9   124.9
124.0   124.5
124.2   124.8
123.9   124.8
123.8   124.5
123.8   124.6
124.0   125.0
123.9   124.7
123.6   124.9
123.5   124.9
123.4   124.7
123.4   124.2
123.4   124.7
123.4   124.8
123.3   124.4
123.3   124.6
123.5   124.4
123.6   124.0
123.8   124.7
123.9   124.4
123.9   124.6
123.8   124.6
123.9   124.6
123.7   124.8
123.8   124.3
123.8   124.0

Another medical student named Emily is also studying the population of pregnant women in the United States, and is also interested in the duration of their pregnancies (in days). Emily will compute a 95% confidence interval. Like Sheena, Emily knows that the population standard deviation equals 16 days. Emily takes a random sample of 20 pregnant women (this is a different random sample than Sheena's!). The sample mean duration among the 20 pregnancies in Emily's sample equals 278. Emily's 95% confidence interval equals ( A, B ). What is the value of B? Round off to the second decimal place.

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Question 71 pts

Suppose Emily decided that the error margin of her 95% confidence interval was too large and wanted an error margin of 1.7 days while maintaining a 95% confidence level. She should take another random sample of size n = _________. (Remember to always round sample sizes up to the next integer.)

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Question 81 pts

If Emily had used the same data to compute a 99% confidence interval (instead of a 95% confidence interval), it would have been _________.

Group of answer choices

the same width

wider

thinner

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Question 91 pts

For both Sheena’s hypothesis test and Emily's confidence interval, what is statement is true?

Group of answer choices

Since the sample size is large, they are NOT required to assume the population is normally distributed.

Since the sample size is large, they ARE required to assume the population is normally distributed.

Since the sample size is small, they are NOT required to assume the population is normally distributed.

Since the sample size is small, they ARE required to assume the population is normally distributed.

We want to examine the efficacy of the current flu vaccine at preventing the flu. Once flu season is over we ask 500 people if they got the vaccine and if they contracted the flu. We then break them into groups (those who got the vaccine and those who did not) and compare them based upon whether or not they contracted the flu.

Suppose your test statistic is statistically significant. Interpret what this significant result means in terms of the alpha and p-value AND what would you conclude about the null hypothesis for this particular research (retain or reject and state in words the conclusion you would draw about the relationship between the two variables).

The amount of corn chips dispensed into a 10-ounce bag by the dispensing machine has been identified at possessing a normal distribution with a mean of 10.5 ounces and a standard deviation of 0.2 ounces (these are the population parameters). Suppose a sample of 100 bags of chips were randomly selected from this dispensing machine. Find the probability that the sample mean weight of these 100 bags exceeded 10.6 ounces. (Hint: think of this in terms of a sampling distribution with sample size = 100)