~Note: This is much more realistic and practical.  The population standard deviation is usually not known.

~Note:  t curves are used instead of the standard normal curve since we are using the sample standard deviation in place of the population standard deviation.

~Note:  There are many t curves, so be sure you select the right one for your test with the proper degree of freedom (n-1)

~Assumptions:  

1) Population standard deviation is not known     

2) either the population is normally distributed or n is 30 or more.

~Test Statistic:   use  t = ( x bar - m_o)/[s/√n], where x bar is the sample mean, m_o is the mean of Ho, s is the sample standard deviation, n is the sample size.

(In the p value method, the TI-83 will display this for you automatically)

~Critical values:  They change, depending on the significance level and the t curve you use. Get these from table A-3 (t distribution)

~Note:  The p value method is best here since the test statistic is calculated automatically & you do not need to get critical values.

~Note:  The procedures are the same as in the previously covered tests, however, t curves are used. (go to T-Test for the TI-83, menu 2, under STAT, TESTS)

~Example:  The average residential energy expenditure for a given year was $1022 per household. That same year, 15 randomly selected upper-income families reported the following energy expenditures: $1153, $1514, $1610, $1249, $1420, $1192, $1126, $807, $1104, $1053, $1130, $1250, $1689, $1268, and $1084.  

At the 5% significance level, do the results of the sample suggest that upper-income families spend more?

Assume that energy expenditures are approximately normal distributed.

~Procedure: Enter data in L1, STAT, TESTS, T-Test (menu 2),
H_o: m = 1022,  List  L1, Freq: 1,  > m_o,  Calculate, Enter.

You will see t=3.71 (test statistic),  p = .0012 (p-value) < .05

~Conclusion:  There is sufficient evidence to conclude that upper-income families spend more. 

~Example:  The bumpers on a new line of cars are supposed to sustain only minor damage in collisions at speeds up to 5 mph.

In a test of 5 of these cars, the mean speed for minor damage was 4.8 mph with a standard deviation of 0.3 mph. Assume that the variable representing the mean speeds is approximately normally distributed.

Are the test results statistically significant (do we reject H₀) at the 0.05 level?

~Procedure: STAT, TESTS, T-Test (menu 2), Stats, m_o= 5, S_x =.3,
n=5,  < m_o, Calculate, Enter.

You will see the test statistic & p-value of .105, which is > .05, so,
we do not reject the null hypothesis.

~Conclusion: There is sufficient evidence to conclude that minor damages occur up to speeds of 5 mph, not less than 5 mph.

~Example:  All women on a special diet have an estimated cholesterol level of 184. These levels are normally distributed.

A sample of 10 female students at Marist on this diet gave the following results: 176, 180, 175, 186, 182, 188, 180, 186, 168, 184.

Many thought this level of 184 for all women is too high. Test this against the data at the 5% level of significance.

~Procedure: Enter the data in a list & use a T-Test. The p-value will be .0523 > .05, so, we do not reject the null hypothesis that the average level = 184.

~Conclusion: There is sufficient evidence to conclude that women on a special diet have a mean cholesterol level of 184, not lower.

OR, There is insufficient evidence to conclude that women on a special diet have a mean cholesterol level less than 184.