Briefly discuss the consequences of type I and type II errors in the context of this problem. From your point of view, which of the two errors is the most serious for this situation?

Question 1: (MINITAB)

50. Generate 25 samples of size 100 from a normal population with mean m=500 and standard deviation s= 50. Do not put this in your report.
    1. Use MINITAB to construct 95% confidence intervals for the population for each of the 25 samples generated in (a). Copy and paste the output into your report.
51. Verify, by hand and for sample 1 only, the results obtained by Minitab.
52. Suppose you were to construct 95% confidence intervals for the population for each of the 25 samples generated in (a). Would all 25 intervals contain the value of the population mean? Explain your answer.
53. How many intervals in (b) failed to capture the true value of the population mean? Mark them on the Minitab output in your report. Is that number what you expected? Explain.

 

Question 2: (by hand)

Lead in amounts over the primary drinking water standard of 0.0150 milligrams per liter may cause nervous system disorders and brain or kidney damage. Since lead accumulates in body tissue, it is especially hazardous to the fetus or to children under three years old. Over time, the standard deviation of the lead concentration in samples of the drinking water of a neighborhood near an old battery plant has been s=0.0005 mg/l. This week, 16 samples of drinking water were collected from homes in the neighborhood with a mean of =0.0153.

1. a) Do these statistics provide sufficient evidence to require residents to stop drinking the water and to incur the expense of having water trucked into the neighborhood? Do a hypothesis test and show all 6 steps. Use .
2. b) Briefly discuss the consequences of type I and type II errors in the context of this problem. From your point of view, which of the two errors is the most serious for this situation?
3. c) Construct a 95% confidence interval for the mean amount of lead in the water.
4. d) What is the power of the test for an alternative mean of 0.0154?

 

Question 3: (by hand)

A biologist estimates that the chance of germination for a type of bean seed is 0.8. A student was given 10 seeds. Let X be the number of seeds germinated from 10 seeds.

1. Assuming that the germination of seeds is independent, explain why the distribution of X is binomial.
2. What are the values of n and p?
3. What is the mean and standard deviation of the number of seeds that germinate?
4. What is the probability that all seeds germinate?
5. What is the probability that only one seed does not germinate?
6. What is the probability that at most four seeds germinate?
7. Suppose that a group of students plant 100 seeds. What is the mean and standard deviation of the number that germinate?
8. What is the probability that at least 70 of the seeds germinate?

 

Question 4: (Excel)

This is a simulation to test how the distribution of the sample mean, its mean and standard deviation relate to their population counterparts as the sample size increases. Your data file Lab2205-152 contains worksheets that generate 100 samples of sizes 4, 9, and 16 respectively from a standard normal distribution and 100 samples of sizes 4, 9, and 16 respectively from a uniform distribution on the interval [0,1] which has a mean of 0.5 and a standard deviation of  = 0.288675. Look carefully at the formulas in the first 6 rows in the worksheets.

1. State theQuestion 1: (MINITAB)
   50. Generate 25 samples of size 100 from a normal population with mean m=500 and standard deviation s= 50. Do not put this in your report.
       1. Use MINITAB to construct 95% confidence intervals for the population for each of the 25 samples generated in (a). Copy and paste the output into your report.
   51. Verify, by hand and for sample 1 only, the results obtained by Minitab.
   52. Suppose you were to construct 95% confidence intervals for the population for each of the 25 samples generated in (a). Would all 25 intervals contain the value of the population mean? Explain your answer.
   53. How many intervals in (b) failed to capture the true value of the population mean? Mark them on the Minitab output in your report. Is that number what you expected? Explain.

    

   Question 2: (by hand)

   Lead in amounts over the primary drinking water standard of 0.0150 milligrams per liter may cause nervous system disorders and brain or kidney damage. Since lead accumulates in body tissue, it is especially hazardous to the fetus or to children under three years old. Over time, the standard deviation of the lead concentration in samples of the drinking water of a neighborhood near an old battery plant has been s=0.0005 mg/l. This week, 16 samples of drinking water were collected from homes in the neighborhood with a mean of =0.0153.

   1. a) Do these statistics provide sufficient evidence to require residents to stop drinking the water and to incur the expense of having water trucked into the neighborhood? Do a hypothesis test and show all 6 steps. Use .
   2. b) Briefly discuss the consequences of type I and type II errors in the context of this problem. From your point of view, which of the two errors is the most serious for this situation?
   3. c) Construct a 95% confidence interval for the mean amount of lead in the water.
   4. d) What is the power of the test for an alternative mean of 0.0154?

    

   Question 3: (by hand)

   A biologist estimates that the chance of germination for a type of bean seed is 0.8. A student was given 10 seeds. Let X be the number of seeds germinated from 10 seeds.

   1. Assuming that the germination of seeds is independent, explain why the distribution of X is binomial.
   2. What are the values of n and p?
   3. What is the mean and standard deviation of the number of seeds that germinate?
   4. What is the probability that all seeds germinate?
   5. What is the probability that only one seed does not germinate?
   6. What is the probability that at most four seeds germinate?
   7. Suppose that a group of students plant 100 seeds. What is the mean and standard deviation of the number that germinate?
   8. What is the probability that at least 70 of the seeds germinate?

    

   Question 4: (Excel)

   This is a simulation to test how the distribution of the sample mean, its mean and standard deviation relate to their population counterparts as the sample size increases. Your data file Lab2205-152 contains worksheets that generate 100 samples of sizes 4, 9, and 16 respectively from a standard normal distribution and 100 samples of sizes 4, 9, and 16 respectively from a uniform distribution on the interval [0,1] which has a mean of 0.5 and a standard deviation of  = 0.288675. Look carefully at the formulas in the first 6 rows in the worksheets.

   1. State the central limit theorem in your own words.
   2. What formula is used to calculate the mean of the distribution of sample means in the spreadsheet? Be sure to tell which rows and columns are involved in the calculation.
   3. What formula is used to calculate the standard deviation of the distribution of sample means in the spreadsheet? Be sure to tell which rows and columns are involved in the calculation.
   4. Look at the worksheet with normally distributed data with sample size 4. Type a 0 in a blank row or column such as cell A3. Copy and paste cells in rows 1 and 2 columns A-D into your lab report. The original mean is 0 and the original standard deviation is 1. Is the mean of the distribution of the distribution of sample means close to the original mean? How is the standard deviation of the distribution of sample means related to the original standard deviation? Is it close to what it should be? Repeat with the uniformly distributed data.
   5. Look at the worksheet with normally distributed data sample size 9. Type a 0 in a blank row or column such as cell A3. Copy and paste cells in rows 1 and 2 columns A-D into your lab report. The original mean is 0 and the original standard deviation is 1. Is the mean of the distribution of the distribution of sample means close to the original mean? How is the standard deviation of the distribution of sample means related to the original standard deviation? Is it close to what it should be? Repeat with the uniformly distributed data.
   6. Look at the worksheet with normally distributed data sample size 16. Type a 0 in a blank row or column such as cell A3. Copy and paste cells in rows 1 and 2 columns A-D into your lab report. The original mean is 0 and the original standard deviation is 1. Is the mean of the distribution of the distribution of sample means close to the original mean? How is the standard deviation of the distribution of sample means related to the original standard deviation? Is it close to what it should be? Repeat with the uniformly distributed data.
   7. Look at the worksheet with normally distributed data sample size 16. Now make histograms of the sample means (column B) and the first sample (column D) in Excel. (Look at the instructions for lab 1 if you can’t remember how.) Describe the similarities and differences between the two histograms. In particular, describe the location of the center for each and the compare their shapes and how spread they are. Repeat for the uniform data with sample size 16.

    

    

   
   Instructions

   Question 1:

   MINITAB: This is a simulation exercise. The data will be generated by the software

   1. a) To simulate 25 samples of size 100 each:
   • Choose Calc → Random Data → Normal
   • Type 100 in the ‘Generate—rows of data’ box and press Tab
   • Type C1-C25 in the ‘Store in column(s)’ box and press Tab
   • Type 500 in the ‘Mean’ box and press Tab
   • Type 50 in the ‘Standard deviation’ box and press enter

   Columns C1-C25 form the 25 samples. Do not copy the data to your Word file.

   1. b) To construct the 95% confidence intervals for each of the 25 samples:
   • Choose Stat → Basic Statistics → 1-Sample z
   • Type C1-C25 in the ‘Samples in columns’ box and press tab
   • Enter the given value ofin the ‘Standard deviation’ box
   • Choose ‘Options’
   • Enter 95 in the ‘ Confidence Level’ box and press Enter twice

   Copy the output to your MS Word file and answer the questions.

    

   Question 4:

    

   For parts (d) through (g):

    

   Part 1:

   1-i) To select 100 random samples of size n=4 and compute the sample means from a normal distribution, choose

   • Distributions → Continuous distributions → Normal distribution→ Sample from normal distribution
   • Enter the population mean and standard deviation  in the Mean and Standard deviation boxes, respectively
   • Enter 100 in the “Number of samples” box
   • Enter the sample size (n=4) in the “Number of observations” box and click OK
   • Click on the View data set button. The 100 samples are the rows (they are labeled as such). Their means are computed and stored in the mean column. No need “to copy and paste” anything at this time

   1-ii) To find the histogram of the 100 sample means:

   • Choose Graphs → Histogram
   • Pick the “mean” variable
   • Click on Options and
   • Select “Percentages” from Axis Scaling
   • Enter Sample means in the x-label box
   • Leave the y-label box empty
   • Insert a title in the Graph title field (for e.g., Approximate Sampling Distribution of the Sample Mean (n=4)) and click OK. Copy and paste to your Word document
   • Click OK. The output appears in a separate graph window

   1-iii) To find the summary statistics:

   • Choose Statistics → Summaries → Numerical summaries
   • Pick the “mean” variable
   • Click on Options and choose Mean and Standard deviation and deselect everything else.
   • Click OK. Copy and paste.

    

   Repeat the 3 steps above for n=9 and then for n=16. Compare the three distributions thus obtained.

   Answer the questions.

    

   Part 2:

   You repeat the above for the uniform distribution on the interval (0,1).

   2-i) To select 100 random samples of size n=4 and compute the sample means from a uniform distribution, choose

   • Distributions → Continuous distributions → Uniform distribution→ Sample from uniform distribution
   • Enter the endpoints of the interval in the Minimum and Maximum boxes, respectively (try the interval (0,1))
   • Enter 100 in the “Number of samples” box
   • Enter the sample size (n=4) in the “Number of observations” box and click OK
   • Click on the View data set button. The 100 samples are the rows (they are labeled as such). Their means are computed and stored in the mean column. No need “to copy and paste” anything at this time

   2-ii) To find the histogram of the 100 sample means:

   • Choose Graphs → Histogram
   • Pick the “mean” variable
   • Click on Options and
   • Select “Percentages” from Axis Scaling
   • Enter Sample means in the x-label box
   • Leave the y-label box empty
   • Insert a title in the Graph title field (for e.g., Approximate Sampling Distribution of the Sample Mean (n=4)) and click OK. Copy and paste to your Word document
   • Click OK. The output appears in a separate graph window

   2-iii) To find the summary statistics:

   • Choose Statistics → Summaries → Numerical summaries
   • Pick the “mean” variable
   • Click on Options and choose Mean and Standard deviation and deselect everything else.
   • Click OK. Copy and paste.

    

   Repeat the 3 steps above for n=9 and then for n=16. Compare the three distributions thus obtained.

2. What formula is used to calculate the mean of the distribution of sample means in the spreadsheet? Be sure to tell which rows and columns are involved in the calculation.
3. What formula is used to calculate the standard deviation of the distribution of sample means in the spreadsheet? Be sure to tell which rows and columns are involved in the calculation.
4. Look at the worksheet with normally distributed data with sample size 4. Type a 0 in a blank row or column such as cell A3. Copy and paste cells in rows 1 and 2 columns A-D into your lab report. The original mean is 0 and the original standard deviation is 1. Is the mean of the distribution of the distribution of sample means close to the original mean? How is the standard deviation of the distribution of sample means related to the original standard deviation? Is it close to what it should be? Repeat with the uniformly distributed data.
5. Look at the worksheet with normally distributed data sample size 9. Type a 0 in a blank row or column such as cell A3. Copy and paste cells in rows 1 and 2 columns A-D into your lab report. The original mean is 0 and the original standard deviation is 1. Is the mean of the distribution of the distribution of sample means close to the original mean? How is the standard deviation of the distribution of sample means related to the original standard deviation? Is it close to what it should be? Repeat with the uniformly distributed data.
6. Look at the worksheet with normally distributed data sample size 16. Type a 0 in a blank row or column such as cell A3. Copy and paste cells in rows 1 and 2 columns A-D into your lab report. The original mean is 0 and the original standard deviation is 1. Is the mean of the distribution of the distribution of sample means close to the original mean? How is the standard deviation of the distribution of sample means related to the original standard deviation? Is it close to what it should be? Repeat with the uniformly distributed data.
7. Look at the worksheet with normally distributed data sample size 16. Now make histograms of the sample means (column B) and the first sample (column D) in Excel. (Look at the instructions for lab 1 if you can’t remember how.) Describe the similarities and differences between the two histograms. In particular, describe the location of the center for each and the compare their shapes and how spread they are. Repeat for the uniform data with sample size 16.

 

 

Instructions

Question 1:

MINITAB: This is a simulation exercise. The data will be generated by the software

1. a) To simulate 25 samples of size 100 each:

• Choose Calc → Random Data → Normal
• Type 100 in the ‘Generate—rows of data’ box and press Tab
• Type C1-C25 in the ‘Store in column(s)’ box and press Tab
• Type 500 in the ‘Mean’ box and press Tab
• Type 50 in the ‘Standard deviation’ box and press enter

Columns C1-C25 form the 25 samples. Do not copy the data to your Word file.

1. b) To construct the 95% confidence intervals for each of the 25 samples:

• Choose Stat → Basic Statistics → 1-Sample z
• Type C1-C25 in the ‘Samples in columns’ box and press tab
• Enter the given value ofin the ‘Standard deviation’ box
• Choose ‘Options’
• Enter 95 in the ‘ Confidence Level’ box and press Enter twice

Copy the output to your MS Word file and answer the questions.

 

Question 4:

 

For parts (d) through (g):

 

Part 1:

1-i) To select 100 random samples of size n=4 and compute the sample means from a normal distribution, choose

• Distributions → Continuous distributions → Normal distribution→ Sample from normal distribution
• Enter the population mean and standard deviation  in the Mean and Standard deviation boxes, respectively
• Enter 100 in the “Number of samples” box
• Enter the sample size (n=4) in the “Number of observations” box and click OK
• Click on the View data set button. The 100 samples are the rows (they are labeled as such). Their means are computed and stored in the mean column. No need “to copy and paste” anything at this time

1-ii) To find the histogram of the 100 sample means:

• Choose Graphs → Histogram
• Pick the “mean” variable
• Click on Options and
• Select “Percentages” from Axis Scaling
• Enter Sample means in the x-label box
• Leave the y-label box empty
• Insert a title in the Graph title field (for e.g., Approximate Sampling Distribution of the Sample Mean (n=4)) and click OK. Copy and paste to your Word document
• Click OK. The output appears in a separate graph window

1-iii) To find the summary statistics:

• Choose Statistics → Summaries → Numerical summaries
• Pick the “mean” variable
• Click on Options and choose Mean and Standard deviation and deselect everything else.
• Click OK. Copy and paste.

 

Repeat the 3 steps above for n=9 and then for n=16. Compare the three distributions thus obtained.

Answer the questions.

 

Part 2:

You repeat the above for the uniform distribution on the interval (0,1).

2-i) To select 100 random samples of size n=4 and compute the sample means from a uniform distribution, choose

• Distributions → Continuous distributions → Uniform distribution→ Sample from uniform distribution
• Enter the endpoints of the interval in the Minimum and Maximum boxes, respectively (try the interval (0,1))
• Enter 100 in the “Number of samples” box
• Enter the sample size (n=4) in the “Number of observations” box and click OK
• Click on the View data set button. The 100 samples are the rows (they are labeled as such). Their means are computed and stored in the mean column. No need “to copy and paste” anything at this time

2-ii) To find the histogram of the 100 sample means:

• Choose Graphs → Histogram
• Pick the “mean” variable
• Click on Options and
• Select “Percentages” from Axis Scaling
• Enter Sample means in the x-label box
• Leave the y-label box empty
• Insert a title in the Graph title field (for e.g., Approximate Sampling Distribution of the Sample Mean (n=4)) and click OK. Copy and paste to your Word document
• Click OK. The output appears in a separate graph window

2-iii) To find the summary statistics:

• Choose Statistics → Summaries → Numerical summaries
• Pick the “mean” variable
• Click on Options and choose Mean and Standard deviation and deselect everything else.
• Click OK. Copy and paste.

 

Repeat the 3 steps above for n=9 and then for n=16. Compare the three distributions thus obtained.