In: Math
Faced with rising fax costs, a firm issued a guideline that transmissions of 8 pages or more should be sent by 2-day mail instead. Exceptions are allowed, but they want the average to be 8 or below. The firm examined 37 randomly chosen fax transmissions during the next year, yielding a sample mean of 10.22 with a standard deviation of 4.88 pages. |
(a-1) |
Find the test statistic. (Round your answer to 4 decimal places.) |
The test statistic |
(a-2) |
At the .01 level of significance, is the true mean greater than 8? |
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(b) |
Use Excel to find the right-tail p-value. (Round your answer to 4 decimal places.) |
p-value |
Before we go on to solve the problem let us know a bit about t-test and p-value.
P-Value
The choice of a specific value of α is completely arbitrary and is determined by non-statistical considerations such as the possible consequences of rejecting H0 falsely and the economic and practical implications of the decision to reject H0. There is another value associated with a statistical test, it is called the probability value or p-value.
Definition: The p-value associated with a test is probability that we obtain the observed value of the test statistic or a value that is more extreme in the direction given by the alternative hypothesis when H0 is true.
Example: Let X~Normal(μ,σ2), to test H0:μ=4 ag. HA:μ>4 if we take a random sample of size n=9 and we are given that,
,
then the observed value of the test statistic is
Then the p-value is,
[Here t8 is t-distribution with 8 degrees of freedom]
Hence, here if α=0.05 we would have reject H0 since p-value is less than 0.05
The smaller the p-value, the more extreme the outcome the outcome and the stronger the evidence against H0.
If α is the chosen level of significance we reject H0 if,
and we accept H0 if,
.
Coming back to our problem,
Given that the firm examined 37 randomly chosen fax transmissions during the next year, yielding a sample mean of 10.22 with a standard deviation of 4.88 pages.
Clearly,
The hypothesis is,
a-1) Here we need to find the test statistic,
The test statistic is given by,
n=37, μ0=8
Hence the test statistic is given by 2.7672.
a-2) Before we solve a-2) let us solve part b)
b) To find the right tail p-value for t-distribution we use the following formula,
where,
t=the value of the test statistic
degFreedom=n-1
tail=1(if using a one tail test) or 2(if using a two tail test)
Here
t=2.7672
degFreedom=37-1=36
tail=1(Since it is a one tail distribution)
Then press enter,
Clearly the p-value is 0.0044.
a-2) Now we here level of significance α=0.01 and p-value is 0.0044
Clearly,
Clearly we reject H0:μ=8 and accept H0:μ>8.
Hence at level of significance α=0.01, Yes the true mean is greater than 8.