5 Exercises 5

Exercise 5.1 A company markets bed legs. For these legs, a plastic ring with an inner diameter $x$ is used. We define a random variable $X$ which associates to each randomly chosen leg the inner diameter of its ring $x$ measured in millimeters. We admit that $X$ follows a normal distribution of mean $\mu$ and standard deviation $0.04$. The supplier affirms that $\mu=12.1 \, mm$.

We have a doubt about this statement. A sample of $64$ legs is taken from the delivery. The average diameter on this sample is $12.095\, mm$. What do we conclude about the average diameter of the rings at significance level of $10\%$?

Exercise 5.2 A factory manufactures cables. A cable is considered “compliant” if its breaking strength is greater than $3$ tons. The engineer in charge of production would like to know, on average, the breaking strength of the manufactured cables.

Of course, it is out of the question to do the breaking strength test on the whole production (the factory would lose all its production!). Let us note $X$ the random variable corresponding to the force to be exerted on the cable to break it (in tons). For this question, we will suppose that $X \thicksim \mathcal{N}(\mu, \sigma^2)$ and that the value of $\sigma$ is known and equal to $0.38$.

A technician takes a sample of 100 cables in the production. With the data from the sample, the technician obtains the following results: the average breaking strength of the 100 cables in the sample is 3.5 tons.

Describe the random experiment, the population, the probability used on the probability space and the random variable studied.
What does the parameter $\mu$ represent? Give an estimator and then an estimate of $\mu$.
Can we say, with a risk of error of $2.5\%$, that the average breaking strength of all the cables in the production is strictly greater than $3$ tons?

2nd part

The proportion of cables with a strength greater than $3$ tons in this sample is $0.85$.

Describe the new random variable $Y$. What is its distribution?
Give a point estimate of the proportion $p$ of compliant cables in the production.
Can we say, with a risk of error of $5\%$, that the proportion $p$ of compliant cables in the production is strictly greater than $0.80$?

Exercise 5.3 A new potato variety is used on a farm. The average yield of the old variety was 41.5 tons per hectare. The new variety is grown on 100 hectares, with an average yield of 45 tons per hectare and a sample’s standard deviation of 11.25.

Should the new variety be encouraged to be grown, with a risk of error of $1\%$?

Exercise 5.4 An article in the journal Growth¹ reported the results of a study that measured the weight (in grams) of guinea pigs at birth:

421	494.6	110.7	102.4	317	447.8	879	273	279.3
452.6	373.8	96.4	241	290.9	687.6	88.8	268	258.5
456.1	90.5	81.7	296	256.5	705.7	296	227.5	296

Test the hypothesis that the average weight is $300$ grams with $95\%$ confidence.
Explain how to answer the previous question with a two-sided CI for the mean weight.

Exercise 5.5 An article in an agricultural journal² determined that the essential amino acid (Lysine) composition of soybean meals are as shown here (g/kg):

22.2

24.7

20.9

24.8

26.5

23.8

25.6

23.9

Can we say, at a risk of $1$%$, that the variance of the quantity of Lysine is equal to $1$?
Confirm the result of the test with a two-sided CI for this variance.

Article entitled “Comparison of Measured and Estimated Fat-Free Weight, Fat, Potassium and Nitrogen of Growing Guinea Pigs”↩︎
Article entitled “Non-Starch Polysaccharides and Broiler Performance on Diets Containing Soyabean Meal as the Sole Protein Concentrate” in the Australian Journal of Agricultural Research Research”↩︎