Predict the expected number of interruptions for a day that has 150 users per hour on average, using a
point estimate and a 95% interval.

DATA four;
INPUT interruptions usage;
cards;
0 104.2
2 124.6
5 176.3
6 169.3
1 104.6
2 115.8
3 127.8
6 179.4
8 210.5
4 126.7
0 100.5
1 119.5
1 123.8
0 106.4
4 156.7
3 148.2
5 156.2
6 167.3
8 198.2
2 124.6
3 145.9
4 156.2
;
run;

Show X(1) (first oder statistics) is complete by definition.

Reformulate your hypothesis test from your week 5 discussion to incorporate a 2-sample hypothesis test, as specified in Chapter 10. What would be your data? What is your null hypothesis? What is your alternate hypothesis? What would be your Type 1 and Type 2 errors relative to your decision? Suppose you have a p-value of 0.01, what does this mean relative to your problem and decision? Suppose your p-value is 0.20, what does this mean relative to your problem and decision? If you reformulated your design for 3 or more samples, what would be the implications of interaction? When would you use the Tukey HSD or the Tukey-Kramer test, and WHY?

1.

a.) Let z be a random variable with a standard normal distribution. Find the indicated probability. (Enter your answer to four decimal places.) P(−2.20 ≤ z ≤ 1.01)  =

b.) Let z be a random variable with a standard normal distribution. Find the indicated probability. (Round your answer to four decimal places.) P(−1.76 ≤ z ≤ −1.17)  =

c.) Assume that x has a normal distribution with the specified mean and standard deviation. Find the indicated probability. (Round your answer to four decimal places.) μ = 15.1; σ = 4.1

P(10 ≤ x ≤ 26) =

d.) Assume that x has a normal distribution with the specified mean and standard deviation. Find the indicated probability. (Round your answer to four decimal places.) μ = 14.6; σ = 3.6

P(8 ≤ x ≤ 12) =

e.) Assume that x has a normal distribution with the specified mean and standard deviation. Find the indicated probability. (Round your answer to four decimal places.) μ = 100; σ = 15

P(x ≥ 120) =

f.) Find z such that 3.0% of the standard normal curve lies to the left of z. (Round your answer to two decimal places.) z =

g.) Find z such that 57% of the standard normal curve lies to the right of z. (Round your answer to two decimal places.) z =

h.) A person's blood glucose level and diabetes are closely related. Let x be a random variable measured in milligrams of glucose per deciliter (1/10 of a liter) of blood. Suppose that after a 12-hour fast, the random variable x will have a distribution that is approximately normal with mean μ = 86 and standard deviation σ = 21. Note: After 50 years of age, both the mean and standard deviation tend to increase. For an adult (under 50) after a 12-hour fast, find the following probabilities. (Round your answers to four decimal places.)

(i) x is more than 60

(ii) x is less than 110

(iii) x is between 60 and 110

(iv) x is greater than 125 (borderline diabetes starts at 125)

i.) Thickness measurements of ancient prehistoric Native American pot shards discovered in a Hopi village are approximately normally distributed, with a mean of 4.6 millimeters (mm) and a standard deviation of 1.5 mm. For a randomly found shard, find the following probabilities. (Round your answers to four decimal places.)

(i) the thickness is less than 3.0 mm
(ii) the thickness is more than 7.0 mm
(iii) the thickness is between 3.0 mm and 7.0 mm

Consider the AR(1) model xt = 0:5x_t-1 + w_t. Derive the partial acf hh when h = 1 and 2. Show your work or provide justification. You need to start from the definition.

What is the probability that Z is less than minus − 0.27 0.27 or greater than the​ mean? The probability that Z is less than minus − 0.27 0.27 or greater than the mean is 0.8936

1. Use columns F and G for the Least-Squares line.

1. Use Excel to make a scatter plot of the dat
2. Adjust the values of the x and y axes so that the data is centered in the plot.
3. Put the trendline on your plot.
4. Put the equation of the trendline on your plot.
5. Put the R² value on your plot.
6. The R value is a measure of how well the data fits a line. What is R? Is R + or - ?
7. Make a screen shot of your final plot. How well do you think the data fits the line? (good fit, moderate fit, marginal fit, no fit)

2. A brand of mints come in various flavors. The company says that it makes the mints in the following proportions.

A bag bought at random has the following number of mints in it.

Determine whether this distribution is consistent with company’s stated proportions.

1. What is the null hypothesis?
2. What is the alternative hypothesis?
3. Enter the observed number of times a flavor comes up in your test bag and the expected number of times that the flavor should come up into the X-squared goodness of fit applet.
4. What is the number of degrees of freedom?
5. What is the p-value? Provide a screen shot of your answer.
6. Using a 95% confidence interval, should you accept or reject the null hypothesis?
7. Does the distribution of flavors in your random bag support or contest the company’s state proportions? (yes or no).

3. This problem is the check to see whether you understand the X-squared test. There are only 2 test columns, so you cannot use the X-squared Goodness of Fit applet from the previous problem as it requires 3 or more test intervals.

You are told that a genetics theory says the ratio of tall:short plants is 3:1. You test this claim by growing 200 plants. You obtain 160 tall plants and 40 short plants. Using a X-squared test, determine whether or not your results supports the tall:short = 3:1 claim.

1. What is the null hypothesis for this test?
2. What is the alternative hypothesis?
3. Fill in the following table.

4. What is the value of X² for this data?
5. What is the number of degrees of freedom?
6. Use the X² calculator to compute p (use the right tail option). Provide a screen shot of your calculation.
7. Does this value of p support the null hypothesis at the 10% significance level? (yes or no and explain using your numbers)

Exercise Group

Sample Size

Sample Mean

Sample Standard Deviation

Low

37

78.40541

11.422345

Moderate

59

74.18644

9.861934

High

14

67.78571

10.990755

MSE = 111.3   F = 5.355         p-value = 0.00607

Bonferroni Modified Alpha = 0.5/3 = 0.0167

Run the Bonferroni procedure to compare each pair of group means.  Compute each pair’s difference in means, standard error (using the MSE found above), p-value, and conclusion of significance or non-significance.

1. The mean of the distribution of potato sack weights is 80 lbs with a standard deviation of 4 lbs. Assume that the distribution approximates a Bell curve.

1. Use the z-table calculator and find the weight which is the 70th percentile. Use value from an are (Show a screen shot for your answer.)
2. What is the z-score for a weight of 75 lbs? z-score = (x-µ)/σ
3. Suppose you pick a single sack at random. What is the probability that the weight will be between 85 and 100 lbs? Use area from a value. (Show a screen shot for your answer.)

Use the central limit theorem to the following questions.

4. Suppose you pick a group of 9 sacks instea What would be the standard deviation of the sample’s (group’s) average using the central limit theorem? (i.e. σ_xbar = σ/)
5. What is the probability that a group of 9 sacks will have an average weight between 90 and 100 lbs? Use the z-table calculator with “area from a value”. (Show a screen shot for your answer.)

2. Use Column A.

We want to test to see whether the data taken from 25 test experiments is consistent with the mean equal to 10.3 (µ = 10.3), or is more consistent with the mean greater than 10.3 (µ > 10.3).

Use Summary 5b, Table 2, Column 1.

1. What is the null hypothesis H_o for our test?
2. What is the alternative hypothesis H_a?
3. What type of tail test will we use? (left tail, right tail, or two tails)?
4. What is the mean of the sample xbar?
5. What is the standard deviation of the sample s?
6. What is the size of the sample n?
7. We going to use a t-statistic. Explain why we are not going to use a z-statistic.
8. Calculate the t-statistic using xbar, µ, n, and s.
9. How many degrees of freedom does this data set have?
10. Use the t-distribution calculator to compute a p-value. Include a screen shot of your answer.
11. We want a 99% confidence level. Based on your value of p, should we accept or reject the null hypothesis?
12. What if we want a 95% confidence level? Based on your value of p, should we accept or reject the null hypothesis?

3. Use Columns C and D for this question.

You are measuring weight loss using the same set of people at different times C and D. You want to know whether there is any difference in the weight between the start of the diet and the end of the diet. Column C gives the weight at the beginning of diet time. Column D gives the weight for the SAME person at the end of the diet time. Since there is data for the same person at different times, we will test whether µ(C-D) <= 0 or µ(C-D) > 0 (meaning the diet did cause weight loss) since we have correlated data (matched pairs).

Use Summary 5b, Table 2, Column 1

1. Make a new series of data samples by letting E = C – D. List your new series of 10 numbers.
2. What is the null hypothesis H₀ ?
3. What is the alternative hypothesis H_a ?
4. What type of tail test are we going to use? (left tail, right tail, two tail)
5. What is the mean xbar of this new sample?
6. What is the standard deviation of the sample s?
7. What is the size of the sample n?
8. How many degrees of freedom does this data set have?
9. What is the t-statistic for this sample?
10. Use the t-distribution calculator to compute a p-value. Show a screen shot of your answer.
11. Based on this value of p and using a 90% confidence level, is there a systematic difference in the weight changes between the start and stop time of the diet? Should we accept or reject the null hypothesis?

1. During recent seasons, Major League Baseball has been criticized for the length of the games. A report indicated that the average game lasts 3 hours and 30 minutes. A random sample of 17 games was checked for the times to completion. The mean game time for this sample was 2.955294 hours and the sample standard deviation was 0.13571 hours. Assume that the game time follows a normal distribution.

Can we conclude that the mean time for a game is less than 3.5 hours? Use the .05 significance level.

1. A major west coast city provides one of the most comprehensive emergency medical services in the world. Operating in a multiple hospital system with approximately 20 mobile medical units, the service goal is to respond to medical emergencies with a mean time of 12 minutes or less.

The response times for a random sample of 40 medical emergencies provided a mean of 13.25 minutes. The population standard deviation (σ) is believed to be 3.2 minutes.

The EMS director wants to perform a hypothesis test and if the data provide enough evidence that the response time is longer than 12 minutes, the director will conclude that the emergency service is not meeting the response goal.

Test the above hypothesis using the .05 level of significance. (Hint: because you know σ, you will use the Z table to find the critical value(s)).

A manufacturer of potato chips would like to know whether its bag filling machine works correctly at the 411.0 gram setting. It is believed that the machine is underfilling the bags. A 41 bag sample had a mean of 404.0 grams. A level of significance of 0.01 will be used. Determine the decision rule. Assume the variance is known to be 784.00.

1. Suppose we wish to find the required sample size to find a 90% confidence interval for the population proportion with the desired margin of error. If there is no rough estimate   of the population proportion, what value should be assumed for  ?

2. An analyst takes a random sample of 25 firms in the telecommunications industry and constructs a confidence interval for the mean return for the prior year. Holding all else constant, if he increased the sample size to 30 firms, how are the standard error of the mean and the width of the confidence interval affected?

3. For a given confidence level and population standard deviation, which of the following is true in the interval estimation of the population mean?

An urn contains 5 white and 8 red balls. Assume that white balls are numbered. Suppose that 3 balls are chosen with replacement from that urn. Let Yi = 1 if if the ith white ball is selected and Yi = 0 otherwise, i = 1,2:

Find the EXPECTED VALUE of Yi given that a) Y2 = 1; b) Y2 = 0.