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%.
N | Mean | Std. Deviation | Std. Error Mean | |
---|---|---|---|---|
Gross Receipts (in $Millions0 | 12 | $9.0000 | $1.7581 | $.5075 |
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 H_{o} and H_{a}
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_{ }
H_{a} : µ is not all equal to 10 in value
The null hypothesis H_{o} 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 H_{a }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 (S_{x-bar}) given as S_{x-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 µ_{o} (=10)
can only be transformed into standardized score using the Student t
probability distribution; the equation for doing so is t =
(= µ_{o})/(S_{x-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
, S_{x-bar}, the computed/observed value of t
(t_{ov}), 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,
S_{x-bar} = .5057, t_{ov} =
-1.970, and v = 11 (SPSS/win labels it as df).
Step 5: Specify the decision rule by relating the computed/observed
t value (t_{ov}) to the critical t
value (t_{cv}) 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 H_{o} if t_{ov}
> t_{cv}; Retain H_{o} if otherwise. Note that t_{cv} = = t_{.01, 11} =
±2.718.
Step 6: Draw valid statistical and administrative conclusions.
a) Statistical Conclusion - Retain H_{o}
since t_{ov} = -1.970 is less than t_{cv} = -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.
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Copyright© 1996, Ebenge Usip, all rights reserved.
Last revised: Wednesday, July 24, 2013.