Question

In: Math

Faced with rising fax costs, a firm issued a guideline that transmissions of 8 pages or...

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?

No
Yes

  

(b)

Use Excel to find the right-tail p-value. (Round your answer to 4 decimal places.)

  

  p-value   

Solutions

Expert Solution

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.


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