It is commonly believed that anyone who tabulates numbers is a statistician.
This is like believing that anyone who owns a scalpel is a surgeon.
 Hooke

Chapter 11

Sec. 11.1
Inference for a Population Mean

The last chapter provided practice finding confidence intervals and carrying out tests of significance in a somewhat unrealistic setting.  We needed the population  which we rarely had and were forced to use the sample s as our estimator.  In reality  and  are rarely known.  So…let’s get more specific.

The conditions for inference about a mean are as before:  SRS and a normal distribution.
For these smaller (n<30) samples ALMOST normal is actually good enough as long as the data are mostly symmetric without multiple peaks or outliers.  Smaller samples are better handled with a t distribution since normality cannot be validated using the Central Limit Theorem.  (Later, rules of thumb for different small samples.)

WE THEN ESTIMATE THE STANDARD DEVIATION OF XBAR BY s/ as before but now we name this value “Standard Error”.

When the standard deviation of a statistic is estimated from the data, the result is called the standard error of the statistic with code name SE.   SO……

SE = s/

Substituting this value for the standard deviation   / does not actually yield a normal distribution but rather a “t distribution”.  In reality the graph of a t distribution is similar to a normal density curve: symmetric about 0, single peaked, and bell-shaped.  The spread is a bit wider giving more probability in the tails and less in the center.  (More variation is present when using s in place of .  As the df* increase the density curve gets closer and closer to N(0,1)).

*A new term must be considered when working with t distributions and that is “degrees of freedom”  df– simply put DEGREES OF FREEDOM = n – 1.  (NOTE:  there is a different t value for different sample sizes since small samples have high variability.)

We will have to adjust to these changes by switching from the use of Table A to determine critical t* values to using Table C (t-distributions) on the inside back cover of the textbook.  Looking at the table we get a clue as to the need for df in determining critical values. 

Example 1:  What is the critical value (t*) for a t distribution with 18 df and .90 probability to the left of t???  Examining the table headings we find the values returned are for the probability to the right of t*…we must adjust for our question.   From the table table we find df = 18 in the column .10 instead of .90 so t* = 1.330

Example 2:  Construct a 95% confidence interval for the mean  of a population based on an SRS of size n = 12.  What critical value t* should be used????  Consult Table C for df = 11 and 95% confidence interval at the bottom of the table.  So  t* = 2.201.  Looking more closely at the table we can see the corresponding critical z value of 1.96 as before.

Calculating a one sample t statistic is similar to finding a
one sample z statistic having this formula

t =( xbar -  )/ s/

 
NOTE:  Sometimes data is summarized by giving xbar and its standard error rather than xbar and s.  The standard error of the mean xbar is abbreviated SEM.

Example 3:  A medical study finds that xbar = 114.9 and s = 9.3 for the seated systolic blood pressure of 27 members of one treatment group.  What is the standard error of the mean?

SE = s/    =  9.3/   =   1.789

Example 4:  Biologists studying the levels of several compounds in shrimp embryos reported their results in a table, with the note, “Values are means + SEM for 3 independent samples.”  The table entry for the compound ATP was .84 + .01.  The researchers made 3 measurements of ATP having xbar of .84.  What was the sample standard deviation s for these measurements?

We know xbar = .84 and SE = .01 and n = 3   so….we substitute and solve for s

 

SE = s/   
.01 = s/    
(.01)( ) = s
(.01)(1.732)   = .01732

GOOD NEWS!

Finding a  one-sample t confidence interval or performing a one sample t test  is analogous to the one sample z confidence interval and test.

The confidence interval for a t statistic is

xbar + t * s/
where t* is the upper critical value

A confidence interval or significance test is called robust
if the confidence level or P-value does not change very much
when the assumptions are violated.

Index