Difficulty formulating $H_0$ and $H_1$ in hypothesis testing.

Question

For example in this example I don't know $H_o$ is whether $\eta>28 $ or $\eta<28 $ ,and in general how should we choose $H_o$? I'm really confused with this,any help would be appreciated.

Answer 1

We assume that the 17 cars are chosen at random from among cars with approximately normally distributed MPG's.

The claim $H_0$ is that the population mean MPG is 28 or better. Typing the data into Minitab statistical software (I hope correctly), we see that the mean of the $n = 17$ tested cars is $\bar X = 27.67,$ just below 28 MPG, with $S = 4.2.$ Is that evidence that the claim is untrue?

Notice that a properly formulated null hypothesis must contain an equal sign. Here $H_0: \mu \ge 28$ against the two-sided alternative $H_a: \mu < 28.$ The value 28 is used in computing the t-statistic for this test.

The answer is that the the data do not give strong evidence against the claim. For such a population with $\mu = 28$ it would not be unusual to get a random sample of 17 cars with $\bar X = 27.67$ or smaller. The probability of that is the P-value = .375. Only if the P-value is smaller than 0.05 would one normally accuse the manufacturer of making a false claim.

Printout from Minitab:

MTB > Onet 'MPG';
SUBC>   Test 28;
SUBC>   Alternative -1.  # denotes left-sided alternative

One-Sample T: MPG 

Test of mu = 28 vs < 28

                                     95% Upper
Variable   N   Mean  StDev  SE Mean      Bound      T      P
MPG       17  27.67   4.20     1.02      29.45  -0.32  0.375

Here is a printout of the data as I typed them, so you can proofread my data entry. Proofreading is always a good idea.

MTB > print 'MPG'

Data Display 

MPG
   19.0   20.0   24.0   25.0   26.0   26.8   27.2   27.5   28.0   28.2   28.4
   29.0   30.0   31.0   32.0   33.3   35.0

Finally, you should make sure you understand how the t-statistic -.32 is computed, and how to use a printed t table (DF = n - 1 = 16) to decide whether to reject $H_0$ at the 5% level of significance.

If you have access to software, you should try this in whatever kind of software is provided for your course. Input and output formats differ for various kinds of software, but all require you to show the null and alternative hypotheses in some way, and all should provide the same basic results.

Note: Below is the same t-test done in R statistical software.

MPG = c(19.0, 20.0, 24.0, 25.0, 26.0, 26.8, 27.2, 27.5, 28.0, 28.2,
       28.4, 29.0, 30.0, 31.0, 32.0, 33.3, 35.0)

t.test(MPG, mu = 28, alternative = "less")

    One Sample t-test

 data:  MPG 
 t = -0.3236, df = 16, p-value = 0.3752
 alternative hypothesis: true mean is less than 28 
 95 percent confidence interval:
      -Inf 29.44757 
 sample estimates:
 mean of x 
  27.67059 

Answer 2

If you assume that the data are independent and normal distributed (as it is reasonable to do, but it should be written somewhere on your sheet), the solution of this exercise is a typical t-test (because the standard deviation $\sigma$ is not given). We (or the seller) claim(s) that the average mileage is at least 28, thus \begin{equation} H_0: \mu \ge \mu_0 = 28. \end{equation} Now, for this situation, the alternative-hypothesis is $H_1: \mu < \mu_0 = 28$ (one-sided), because we want to test if our claim $H_0: \mu \ge \mu_0= 28$ is reasonable, which means that we reject our null-hypothesis $H_0$ if the data gives a value too smaller than $28$.