Question

An experiment is being planned to compare the effects of several diets on the weight gain of beef cattle, measured over a 140-day test period. In order to have enough precision to compare the diets, it is desired that the standard error of the mean for each diet should not exceed 5kg. (a) If the population standard deviation of weight gain is guessed to be about 20kg on any of the diets, how many cattle should be put on each diet in order to achieve a sufficiently small standard error? (b) If the guess of the s.d is doubled, to 40kg, does the required number of cattle double? Explain

Answer 1

Part (A) 16 cattle should be put on each diet in order to achieve a sufficiently small standard error.

Part (B) No, the required number of cattle should not double.

Explanation:

Consider the provided information.

The standard error of the mean for each diet should not exceed 5kg.

Therefore, sample mean is =
`\sigma_(\bar x)\leq 5`

Part (A)

The population standard deviation of weight gain is guessed to be about 20kg on any of the diets,

From the above
`\sigma=20`

It is given that
`\sigma_(\bar x)\leq 5`

As we know
`\sigma_(\bar x)=\text{sd(sample mean)} = \frac{\text{(population standard deviation)}}{√(n)}`

Thus,

`(20)/(√(n))\leq 5`

`(20)/(5)\leq √(n)`

`4\leq √(n)`

`16\leq n`

16 cattle should be put on each diet in order to achieve a sufficiently small standard error.

Part (B) If the guess of the s.d is doubled, to 40kg, does the required number of cattle double?

From the above
`\sigma=40`

It is given that
`\sigma_(\bar x)\leq 5`

Thus,

`(40)/(√(n))\leq 5`

`(40)/(5)\leq √(n)`

`8\leq √(n)`

`64\leq n`

64 cattle are required, which is not the double of 16.

Hence, No, the required number of cattle should not double.