Math 110 Chapter 6 by S44y04


									                                  Math 110 Chapter 7
                                      Section 7.1
                         Inference for the Mean of a Population

Inference when  is unknown
In chapter 6, we made the unrealistic assumption that we knew , the population standard
deviation. In practice,  is usually unknown and we must use the standard deviation of the
sample, s, to estimate . This changes some details of tests and confidence intervals. Since
we don't know , we can't calculate the standard deviation of x as              . Instead, we
estimate it using the standard error: The standard error of the sample mean x is:           s        .

When we pretended to know , we based confidence intervals and hypothesis tests for  on
                                 x -
the standardized sample mean z       . The z-statistic has the standard normal
distribution. When we substitute standard error of x for its standard deviation, the
statistic that results does not have a normal distribution. It has a t-distribution. The t-
distribution is bell-shaped, like the normal distribution, but the tails are slightly thicker.
There is a different t-distribution for each sample size. We specify a particular t-
distribution by giving its degrees of freedom = n - 1.
Calculating a t statistic:
Draw a random sample of size n from a population that has the normal distribution with
mean  and standard deviation . The one-sample t statistic

             x 
        t           has the t distribution with n - 1 degrees of freedom.
             s   n

1. A vending machine dispenses servings of hot chocolate. You buy a sample of 4 servings
   from the vending machine and measure each serving to obtain the following data (in
   ml's): 247 246 250 245 . The sample mean x is 247 (verify).
   a) Find the standard error of the mean:

    b) What are the degrees of freedom of the t statistic associated with x ?

2. From Table C, find the indicated t* values:
   a) From the t distribution with 19 degrees of freedom, find the t* having .025
      probability to its right.

    b) From the t distribution with 7 degrees of freedom, find the t* having .90
       probability to its left.

Math 120 Lecture Notes                                                                           62
Confidence Intervals when  is unknown
A confidence level C confidence interval for an unknown  is :
                                          x t*

Exercise: A sample of 5 candies was taken from a bulk food bin. The weights of the
candies are: 4.55 4.60 4.41 4.77 4.23
a) Calculate an 80% confidence interval for the population mean weight .

b) Calculate a 98% confidence interval for the population mean weight .

c) Do you have good evidence that the mean weight of the candies is greater than 4.0

i) State the null and alternate hypothesis

ii) Calculate the test statistic, t.

iii) Find t* values from table C that bracket the t statistic, hence find an interval that
contains the p-value.

    iv)     Decide whether to reject H0

Math 120 Lecture Notes                                                                       63
Exercise: Let x = amount of time for a worker in a clothing manufacturing company to join a
collar to a shirt. A sample of eight workers gave the following figures for the amount of
time (in minutes) needed to join a collar to a shirt:
10    12    13    9   8 14      10    11
Is there enough evidence to conclude that the true mean time to join a collar is more than
10 minutes? (Assume the x's are normally distributed).

a) Compute the sample mean x and the sample standard deviation s. State the degrees of

x = __________              s   = __________         degrees of freedom = _____

b) State the null and alternate hypotheses

c) Compute the test statistic t, and mark it on the horizontal axis of the graph.

d) Locate the two critical values t* from Table C that bracket t, and mark them on the

e) Between what two values does the p-value of the test fall? Is the t value computed in
part c) significant at the  = .10 level? At the  = .15 level?

                                                           t distribution with 7 degrees of freedom

Math 120 Lecture Notes                                                                          64
Exercise: A study on the amount of dye needed to get the best colour for a certain type of
fabric was conducted. The following data give the photometer readings for the colour
density of the fabric when the amount of dye used was 1% of the weight of the fabric.
13.2 11.5 12.9 13.0 11.7 10.4 12.1 12.1 11.5 10.3 11.7 12.3 11.2
Do the data demonstrate that the true mean photometer reading for the colour density is
significantly different from 12? Use 10 percent as the level of significance. (Assume a
normal distribution)
a) Compute the sample mean x and the sample standard deviation s. State the degrees of

x = __________              s   = __________         degrees of freedom = _____

b) State the null and alternate hypotheses

c) Compute the test statistic t, and mark it on the horizontal axis of the graph.

d) From table C, find the t* value associate with  = .10 for a two tailed test. Mark it on
the graph.

e) Do you have enough evidence to reject H0 at the 10% level of significance?

                                                        t distribution with 12 degrees of freedom

Math 120 Lecture Notes                                                                              65
Matched Pairs t Procedure
One common experiment design is to compare the response to two treatments given to the
same subjects, or to make before and after observations on the same subject. We can
determine whether or not there is a significant difference between two treatments by
taking differences of the two observations. If there is not any systematic difference in
the response to the two treatments, we would expect the mean of the differences to be

Example: We would like to know whether most people prefer Brand P (Schweppes) or Brand
Q (Western Family) raspberry ginger ale. A Math 120 class was given a sample of each
brand and were asked to rate the brands The cups were labeled P and Q. After the
students tasted both, they gave each drink a score on scale of 1 to 10. The higher the
score, the better they liked the drink.

  Score for       Score for       Differences
                                                       a) Calculate the differences
 Brand P (x1)    Brand Q (x2)       (x1 - x2)
                                                       x1 - x2
      7                  4
      10                 4
                                                       b) Find the sample mean x of
      4                  2
                                                       the differences
      7                  8
      5                   1                            ___________________
      10                 5
      5                  3                             c) Find the sample standard
      4                  7                             deviations of the differences,
      8                  10                            s
      4                   1
      7                  2                             ____________________
      6                  2
      5                  5

a)   Find a 95% confidence interval for the unknown parameter .

b) Do you believe that there is enough evidence to conclude that most people like Brand P
better than Brand Q ? (use  = 0.05 level of significance)

Math 120 Lecture Notes                                                                  66
Example: Right-handed subjects were tested for reaction times with their left and right
hands. The results (in thousandths of a second) are given below.

Subject:   A     B       C     D     E     F     G     H     I     J     K     L     M     N
Right:     191   97      116   165   116   129   171   155   112   102   188   158   121   133
Left:      224   171     191   207   196   165   177   165   140   188   155   219   177   174

a) Use a 0.05 level of significance to test the claim that there is a difference between the
mean of the right- and left-hand reaction times.

b) Construct a 95% confidence interval to estimate the mean of the differences.

Math 120 Lecture Notes                                                                     67
                                                Math 120
                                               Section 7.2
                                           Comparing Two Means

Often it is useful to compare two means from two different populations. To do this, we
take a random sample from each population, and calculate a separate x and s for each
sample. The two samples should be independent--that is, one sample has no influence on
the other. (this is not the same as matched pairs testing, were we took the differences
then found x and s for the differences.)
If populations are normally distributed, then the following formulas may be used:

Confidence interval for the difference of two means 1 - 2:

                    2        2
                   s1       s2
(x1  x2 )  t *                (degrees of freedom = smaller of n1 – 1 , n2 – 1)
                   n1       n2

To test the hypothesis H0: 1 = 2 , compute the two-sample t test statistic
     x1  x 2
t             (degrees of freedom = smaller of n1 – 1 , n2 – 1)
       2     2
     s1    s2
      n1 n 2

The heights (measured in inches) of 20 randomly selected women and 30 randomly selected
men were independently obtained from the student from the college population. The sample
information is given below. Assume that the heights are approximately normally distributed
for both populations. Find a 95% confidence interval for the difference between the mean
heights: 1 - 2.

Male:      n1 = 30          x 1 = 69.8     s1 = 1.92
Female:    n2 = 20          x 2 = 63.8     s2 = 2.18

Math 120 Lecture Notes                                                                    68
Many students have complained that the soft-drink vending machine A dispenses less drink
than machine B. To test this belief, a student randomly samples several servings from each
machine and carefully measured them, with the results as shown:

Machine A:     n1 = 10   x 1 = 5.38    s1 = 1.59
Machine B:     n2 = 12   x 2 = 5.92    s2 = 0.83

Does this evidence support the hypothesis that the mean amount dispensed by Machine A is
less than the amount dispensed by B? Assume the amounts dispensed by both machines are
normally distributed and complete the test using  = 0.05.

Math 120 Lecture Notes                                                                 69

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