﻿ Hypothesis Testing: One-Sample T-Test about U, Dr. Usip, Economics

# A One-Sample T-Test about Population Mean, U

Problem Description and Data
A successful tax advisor is analyzing a motion picture tax shelter for possible investment. The prospectus claims that movies in the same general category as the one under consideration have grossed an average of \$10,000,000 each in the past. This is an optional level of receipts for the current film (gross receipts less than this would result in an inadequate return on investment, and gross receipts more that this would result in taxable income, which is not the primary objective of the tax shelter). The advisor has taken a random sample of similar films and has complied the following statistics to test the claim of the prospectus (Hamburg & Young, 1994, p. 353):

 Film 1 2 3 4 5 6 7 8 9 10 11 12 Gross Receipts (\$millions) 11.5 7 9 10 10 12 8.5 9 8 6 7.5 9

Question: Formulate an appropriate test using alpha = .02?

Data Entry
Note that Gross receipt is the variable of interest (X); it is a continuous variable. To enter the values, double-click on var in column one; this action opens the Define Variable window. Type receipts in the Variable Name box; then type Gross Receipt as the Variable label box. Next, open the Type window and set Decimal Places to one ( i.e.; type 1 to replace the default value of 2). Finally, open the Labels window and type Gross Receipts in the Variable Label box. Click on Continue option and then Okay to return to the data entry screen.

Notes:
(1) Before executing the procedure according to the Command Sequence stated earlier, the test value must be set equal to 10 ( the hypothesized value of µ).
(2) Because the data are quantitative, the variable Type is automatically set to Numeric.
(3) Select FILE/PRINT or the Printer Icon to send your output to the local printer.

Discussion of the Outputs/Results and Testing Procedure
A. The Outputs/Results
The first table of the output contains summary statistics for the gross receipts. The second table contains all the statistics that are needed to perform the test. Notice that SPSS/win actually incorporates the null statement into the output: Test Value = 10. Thus , it important to check the Test Value box as explained earlier in the command sequence. notice also the interval estimate for the difference is for 98 % which was set under OPTION so that the results are consistent with the significance level of 2%.

One-Sample Statistics

N Mean Std. Deviation Std. Error Mean
Gross Receipts (in \$Millions0 12 \$9.0000 \$1.7581 \$.5075

One-Sample Test

Test Value = 10
t df Sig. (2-tailed) Mean Difference 98% Confidence Interval of the Difference
Lower Upper
Gross Receipts (in
\$Millions)

-1.970

11 .074 -\$1.0000 -\$2.3795 \$.3795

B. The Testing Procedure
Step 1
: State Ho and Ha such that they contradict each other completely or rather relate in a mutually exclusive manner. For this problem, this implies the following statements:
Ho: µ = 10
Ha : µ is not all equal to 10 in value

The null hypothesis Ho reflects the claim of the prospectus that the motion picture will gross exactly \$10 million, on average, as other movies in the same general category. It is the maintained hypothesis under investigation. The alternative Ha says that the average amount the movie could gross is not equal to \$10; it could be more or less than that value. Hence the test is a two-tailed test

Step 2: Specify the level of significance, which in this case is given to you (alpha = .02).

Step 3: Identify the test statistic and its sampling distribution. As explained in class and stated above, the test statistic is the sample mean (
) since the test concerns the parameter µ. Assuming that gross receipts are distributed as normal then according to the Central Limit Theorem (CLT) is also distributed as normal with the expected value E() = µ and the standard error (Sx-bar) given as Sx-bar = S/ (Square root of n), where 'S' is the sample standard deviation. Because the sample size (n) is small, the deviation of from the hypothesized value of µo (=10) can only be transformed into standardized score using the Student t probability distribution; the equation for doing so is t = (= µo)/(Sx-bar). Hence, the test is appropriately referred to as the t-test. It is a one-sample test because the inference concerns the value of only one parameter µ and from one population of data ( i.e.; the gross receipts from the universe of similar films).

Step 4: In practice, this is the juncture that requires sampling from the target population using the appropriate random sampling technique. Also, it is at this step that all the number crunching is done using the computer that runs a statistical program (in our case, SPSS/win). The needed statistics from the computer output are the value of
, Sx-bar, the computed/observed value of t (tov), and the number of degrees of freedom (v) given as v = n-1 for this problem. The SPSS/win outputs presented above contains all these information as follows: = 9.000, Sx-bar = .5057, tov = -1.970, and v = 11 (SPSS/win labels it as df).

Step 5: Specify the decision rule by relating the computed/observed t value (tov) to the critical t value (tcv) that you obtain from the t-table using the values of alpha = .02, and v =11. The rule is stated in the following manner: Reject Ho if tov > tcv; Retain Ho if otherwise. Note that tcv = = t.01, 11 = ±2.718.

Step 6: Draw valid statistical and administrative conclusions.
a) Statistical Conclusion - Retain Ho since tov = -1.970 is less than tcv = -2.218 in absolute value.
b) Administrative Conclusion - Based on the sample evidence, it is reasonable to conclude that the data is consistent with the claim that the mean gross receipts of this type of movie is \$10 million. Thus, the prospectus should take comfort in the fact that there would be no loss of investment since the gross receipts would be just adequate, and there would be no taxable income (which is the primary objective of the tax shelter) since the gross receipts would not be greater than \$10 million.