Question

A producer of soap dispensers claims its machine dispense liquid with a standard deviation of 3.1 ounces. A random sample of 20 machines found the standard deviation to be 4.6 ounces. Is there enough evidence, at the α = 0.05 level of significance, to suggest the standard deviation is not 3.1 ounces?

A manufacturer of 10cm marbles claims the standard deviation of it’s marble's diameter is less than 0.21cm. A random sample of 12 marbles found the standard deviation of the diameter to be 0.14cm. Is there enough evidence, at the α = 0.01 level of significance, to back up the manufacturers claim?

A powerpack is manufactured with a mean lifetime of 4 years and a standard deviation of 0.6 years. What percent of the powerpacks will have a lifetime greater than 4.1 years?

If a company makes LED lights with a mean lifetime of 2000 hours and a standard deviation of 120 hours. What percent of the lights have a lifetime between 1840 and 2100 hours?

Answer 1

1. We have given here that

We need to assess whether there is enough evidence to suggest that the standard deviation is not 3.1 ounces.

Null Hypothesis:

Level of significance

Test statistic: follows a with n-1 df.

Critical value of  . Since it is a two tailed alternative, the critical values are:

or

For our test we have,

The Critical values are or

Since the calculated value of is >, we reject the null hypothesis.

and hence there enough evidence, at the α = 0.05 level of significance, to suggest the standard deviation is not 3.1 ounces.

2.A manufacturer of 10cm marbles claims the standard deviation of it’s marble's diameter is less than 0.21cm. A random sample of 12 marbles found the standard deviation of the diameter to be 0.14cm. Is there enough evidence, at the α = 0.01 level of significance, to back up the manufacturers claim?

We have given here that

We need to assess whether there is enough evidence to suggest that the standard deviation is not 3.1 ounces.

Null Hypothesis:

Level of significance

Test statistic: follows a with n-1 df.

Critical value of  . Since it is a two tailed alternative, the critical values are:

The critical value . Since the calculated value of   , we do not reject the null hypothesis. Hence, there is not enough evidence, at the α = 0.01 level of significance, to back up the manufacturers claim of the marble's standard deviation less than 0.21 cm.

3.A powerpack is manufactured with a mean lifetime of 4 years and a standard deviation of 0.6 years. What percent of the powerpacks will have a lifetime greater than 4.1 years?

It is given that

W need to find out the percent of powerpacks that will have life time>4.1 years.

Let us assume that the lifetime of powerpacks follows a Normal distribution. If we call X as the life time of any powerpack, then we need to find out P(X>4.1).

Since X follows N(4, 0.6^2) we know that

will have N(0,1).

Now for X=4.1, we have

We shall look into a Standard Normal table and find out the proportion

which is 0.4338. Therefore we have 43.38% of  powerpacks will have a lifetime greater than 4.1 years.

4.If a company makes LED lights with a mean lifetime of 2000 hours and a standard deviation of 120 hours. What percent of the lights have a lifetime between 1840 and 2100 hours?

Let us assume that the lifetime of LED lights follow a Normal distribution.

It is given that . Here we need to find out the percent of lights having lifetime between 1840 and 2100 hours.

As before, We need to find

when X=1840,

  

When X=2100,

  

   by the properties of the distribution.

  

Therefore, the percent of the lights have a lifetime between 1840 and 2100 hours=70.65%