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The annual profit for your organisation last year was £1203, per employee. The average for the sector was £1228 per employee. You need to know if there is evidence that your organisation is not as profitable, on average, as your competitors. The standard deviation for the sector is £104 per employee.

Questions:

  1. State $H_0$ and $H_1$ for the test to check

    Assuming that annual per employee profit follows a Normal distribution, the resulting P-value for the test is, P= 0.405 (3 d.p.):

  2. Based entirely on this P-value, what would you conclude from the test?

  3. What is the probability of getting a profit per employee  of £1203, or less, when there is no difference between that for your  company and the industry average?

  4. If your annual per employee profit was such that the test had resulted in a P-value given by P=0.032 and you were testing at  the α=0.01 significance level, what would you conclude and why?

  5. If you chose to reject $H_0$ but $H_0$ was actually true, would you be making a Type I or Type II error?

  6. What assumptions have you made about the annual per employee profit in this test?

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    $\begingroup$ This looks like homework - if it is, could you please add the 'homework' tag? You'll still get an answer, but people will approach it a bit differently to maximise learning. $\endgroup$
    – Peter Ellis
    Commented Sep 9, 2012 at 9:01
  • $\begingroup$ Q1 : So H0 is 1203 and H1>1203 ?? $\endgroup$
    – user14099
    Commented Sep 14, 2012 at 17:50
  • $\begingroup$ So Question 3 : I dont calculate the z value ? My ans was : 1203-1228 = -25 -25/104 = -0.240 Looking at table of the standard normal distribution for z = -0.24 it will be .40517 this mean 40.5% That is, assuming for an average profit per employee of 1228, there is a 40.5% chance of selecting at random an organization with profits of 1203 or less. Is that correct ?? $\endgroup$
    – user14099
    Commented Sep 17, 2012 at 16:18
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    $\begingroup$ Note that if you read the questions very (too?) precisely they don't make sense. Given the question above you know the average profit per employee and you know the average of the sector. There is really nothing left to test, since you know the relevant population parameters instead of just the sample parameters from which you have to infer population parameters. $\endgroup$
    – Erik
    Commented Oct 18, 2012 at 7:16
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    $\begingroup$ This problem has a flaw in it. Hypothesis testing uses a sampling distribution of a sample statistic, and hence a standard error that is the population standard deviation divided by the square root of the sample size. There is no such info here. Otherwise, what you say is correct. $\endgroup$
    – user17623
    Commented Dec 9, 2012 at 7:48

1 Answer 1

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This is a very basic homework question testing the basic concepts of hypothesis testing from a introductory statistics course text. As it is clearly homework I will sketch how to answer the questions.

  1. The null hypothesis is that your company's profits have the same value as the average of the other company's. This tests whether or not a random observation from your company's distribution of profits can reasonably be considered to come from the distribution of profits for the competitors (when the competitors' distribution is assumed to be normal with mean 1228 and standard deviation 104. What should the alternative be, one-sided or two?

  2. Since the p-value is high you can't reject the null hypothesis. How would you state this in a conclusion?

  3. Form the z statistic (1203-1228)/104. Look at the table of the statndard normal distribution and determine the probability that the standard normal is less than or equal to the z value that you computed.

  4. P is greater than the significance level 0.01, so you cannot reject.

  5. Falsely rejecting the null hypothesis is called a Type I error.

  6. Since you compare to a standard normal by subtracting the mean and dividing by the standard deviation of a normal distribution for the competitors what does the null hypothesis say about the distribution for your profits?

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  • $\begingroup$ Michael, where can I send you my feedback. $\endgroup$
    – Joe
    Commented Sep 9, 2012 at 14:39
  • $\begingroup$ Right here, Joe: comments are for obtaining clarifications to answers and questions. $\endgroup$
    – whuber ♦
    Commented Sep 10, 2012 at 16:06
  • $\begingroup$ Question 1 H1: u < 1228 and H0: u >= 1228 $\endgroup$
    – Joe
    Commented Sep 15, 2012 at 17:29
  • $\begingroup$ Question 2. The lower the p-value the stronger the evidence against H0. P value = 0.405 given Significance level is .05 if we assume a 95% confidence level at a normal distribution. P value is higher then the significance level or alpha value thus accept H0 Thus our company is at least just as profitable as the industry standard of £ 1228 suggest. $\endgroup$
    – Joe
    Commented Sep 15, 2012 at 17:30
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    $\begingroup$ We never like to say that we accept the null hypothesis. It is more appropriate to say that you can't reject it. Your statement that your company is at least as profitable as the industry standard is too strong. I would say that you don't have sufficient evidence to conclude that you are less profitable. $\endgroup$
    – Michael R. Chernick
    Commented Sep 15, 2012 at 19:28

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