Names:                                          :
Confidence, Significance and P-Values
Work in groups of 2 or 3 (turn in only one paper per group) and each group MUST have a TI-83 calculator.
  1. Acute myeloblastic leukemia is among the most deadly of cancers. Experience indicates that the time in months that a patient survives after initial diagnosis of the disease is normally distributed with a mean of 13.0 months and a standard deviation of three months. A new treatment is being investigated which should prolong the average survival time without affecting the variability. The survival times for 16 patients receiving the new treatment are listed below.
17.8 12.5 16.7 17.3 12.5 18.3 10.5 13.3
21.4 16.2 10.3 11.8 8.8 13.2 12.5 15.6
  1. Set up a significance test for the information above by defining the null hypothesis H0 and the alternative hypothesis Ha.

    2.  

     
     
     
     
     

  2. Calculate the mean, x-bar, of the sixteen survival times and the actual standard deviation (use Sx not sx) of the data in the table. The value of this standard deviation will be used only in the next question.

    3.  

     
     
     

  3. The problem description states that the standard deviation should be about 3, was this the case?

    4.  

     
     
     

  4. Using the problem description's value of 3 for the population standard deviation, what is the standard deviation of the sampling distribution for x-bar (assuming the null hypothesis is true)?

    5.  

     
     
     

  5. Calculate the P-value of your significance test for the observed value of x-bar.

    6.  

     
     
     

  6. Based on your P-value, do you think that the observed value of x-bar would lie in a 95% confidence interval (for the statistic x-bar = mean of 16 survival times) centered at the population mean µ = 13?





  7. Find a 95% confidence interval centered at 13 for the statistic x-bar. Is the observed value of x-bar in this interval?

 
 

Now we'll repeat the exercises above with some new features of the TI-83 and explore the claim that averages are less variable than individual values and averages of larger samples are less variable than averages of smaller samples. Most likely you already calculated the values of x-bar and the actual standard deviation of the 16 survival times using the 1-VarStats feature of your calculator. Be sure that the 16 times are stored in the list L1 and delete or turn off all Y-variable plots and STAT plots. We can calculate the P-value of a significance or hypothesis test quite quickly with the TI-83. This can be done either with data (such as the list of survival times), or with statistics (just the values of x-bar, and n = sample size). For both methods you'll need to know the population mean, µ, and the population standard deviation, s. Press [STAT] and then select the [TESTS] submenu. Hypothesis tests (of this type anyway) are called Z-Tests on the TI-83 and they correspond to the first choice on the [TESTS] submenu. Select 1:Z-TEST. The hypothesis test editor screen should appear. You move around with the arrow keys and make selections by pressing [ENTER]. Select DATA for the input type. Enter the population mean (13) for µ0 and the population standard deviation (3) for s. Give L1 as the list and set the frequency to 1. The second to last line is where you select the form of the test. You can chose from

a two tailed test 
a left tailed test 
or a right tailed test 
Select the right tail test icon and then select DRAW on the bottom line to see a plot and the P-value. The calculator plots the standardized version of the problem and reports the standardized z-value as well.
What is the z-value reported?
 

Did you get the same P-value as above?
 

Now, go back to the Z-Test screen and select STATS this time for the input type. The calculator should display the screen below.

The values from your last test should remain in the Z-Test editor. Move to the bottom line and select Calculate this time. You should get the same P-value and the same Z-value as before, just no graph this time. The calculate option is nice if you only need a number for the p-value or z-value.

We want to see what happens to the p-values as we increase the sample size n. Assume that instead of an average of 16 survival times, the value of x-bar (roughly 14.3) came from a sample of size n. Record the associated p-values in the table below (Use the Z-Test and STATS option for each value of n).
 
 

p-values for x-bar = 14.29, Pop std dev = 3 different n
n 25 50 75 100
p-value        

You should have seen the Z-values getting bigger (more rare) and the p-values getting smaller. This is because averages are less spread out than individual values and averages of larger samples are less spread out than averages from smaller samples.

Finally we want to see evidence that the standard deviation of a mean of n values, from a population whose standard deviation is s, is well approximated by s / square-root(n). The problem is that in order to check this property out, we need many different samples of the same size, from some big population with a known standard deviation. We'll use the TI-83 to generate these samples. You calculated the standard deviation of the original 16 survival times in part (b) of question 1. You should have gotten a value close to 3 for this standard deviation. Those survival times came from the TI-83 command:

randNorm(13.8,3,16).
The randNorm command is on the [MATH] [PRB] menu (option 7). The first value, 13.8 is the mean of a normal distribution. The second value, 3, is the standard deviation and the last number, 16 (which is optional with default value 1), is the sample size, or the number of random normally distributed values we want. The command above therefore instructs the calculator to generate 16 random values from a population which is normal with mean 13.8 and standard deviation 3. You won't need to use this command other than today (unless you feel like seeing some random normal values in your spare time).

We can therefore issue the command

randNorm(13.8,3,16)[STO >] L1
to store a sample of size 16 in the list L1. We can then run the 1-VarStats command and find the mean, x-bar, of these 16 values.
  1. Store a big number in the variable rand ([MATH][PRB][1]) and then calculate 2 such values of x-bar as explained above. Record your x-bar values on the board, and copy all of the x-bar values from the class below. Round to 3 digits after the decimal point.













  2. The theoretical value of the standard deviation for the sampling distribution of x-bar values is 3 / squareroot(16) = 0.75. Enter the classes data into your calculator and calculate the standard deviation (use Sx not sx) of these x-bar values. Did you get a number close to 0.75?






  3. What was the mean value of the x-bar values?





Finally, the TI-83 can do confidence intervals as well. The confidence interval editor screen is very similar to the hypothesis test screen used above. You need to enter the population standard deviation, plus either a list where your data is stored, or a value for the sample size n and a value for x-bar. There is a line for entering the confidence level as well (as in .95 for a 95% confidence interval).

  1. The last part of question 1 asked for a 95% confidence interval centered at 13 for x-bar = mean of 16 values from a population with standard deviation equal to 3. Use the Z-Interval command ([STAT][TESTS][7]) to check your work. Record the output from this command below.






  2. The class data represents many values of x-bar. Theory predicts that the x-bar values should have a mean equal to 13.8 (the first argument to the randNorm command). Calculate a 70% confidence interval for x-bar values which are means of samples of size 16 from a population with standard deviation 3. Center this interval at 13.8.






  3. List all of the class values which lie outside the 70% confidence interval above.