Recitation 10

P-values, Hypothesis Testing

1.

Cobra Cheese Company buys milk from several suppliers. Cobra suspects that some suppliers are adding water to their milk to increase their profits. Excess water can be detected by measuring the freezing point of the milk. The freezing temperature of natural milk varies normally, with mean $\mu=-0.545$ degrees Celsius and standard deviation $\sigma=0.008$ degrees Celsius. Added water raises the freezing temperature towards 0 degrees Celsius, the freezing temperature of water. Cobra's laboratory manager measures the freezing temperature of 5 consecutive lots of milk from one producer. The mean measurement is $\bar{x} = -0.538$ degrees Celsius. Is this good evidence that this producer is adding water to the milk? State hypotheses, carry out the test, give the P-value, and state your conclusions. The above question is 6.43 from the book. To get full credit for a question such as this you should be sure to state all your assumptions, define your terms, state hypotheses carefully and write sentences to state your conclusions. The following exercise has broken down these steps for this particular example.

(a)

Carefully state a null and alternative hypothesis and define all the terms you use (in this case explain carefully what the $\mu$ is in your hypotheses, also define $\sigma$). (Hint: do the values relate to all producers or one producer?)

(b)

Calculate the observed value of your test statistic z and carefully state the distribution of the test statistic assuming the null hypothesis is true.

(c)

Explain how to calculate the P-value for the test. Calculate it (use tables).

(d)

Interpret your findings. Be sure to write in sentences in the context of the question being asked. (Hint: does your conclusion relate to all producers or one producer).

2.

To illustrate the idea of fixed level testing, do the following problem related to the previous question.

(a)

Consider the exact situation of the previous problem with a one-sided test with n=5 and the null and alternative hypotheses you gave in (a) and the value of $\sigma$ given. For a test of level $\alpha=0.05$ for what values of your test statistic z would you ``reject the null hypothesis'' and conclude in favor of the alternative?

(b)

Re-express the value given in (a) to a value of $\bar{x}$. That is for what values of $\bar{x}$ would you reject the null hypothesis? (Use the values of the null hypothesis and the population standard deviation given in the previous problem).

(c)

Consider increasing the sample size to n=25; would you expect the value in (b) to increase or decrease? (Take a guess and then repeat (a) and (b) above to check your guess).

(d)

Repeat (a) and (b) for a two sided alternative and n=5. That is for what z values would you reject the null hypothesis for a two sided test of level $\alpha=0.05$? To what values of $\bar{x}$ does this correspond?