Question

A manufacturer claims that the mean lifetime,U , of its light bulbs is 54 months. The standard deviation of these lifetimes is 8 months. One hundred fifty bulbs are selected at random, and their mean lifetime is found to be 53 months. Can we conclude, at the 0.05 level of significance, that the mean lifetime of light bulbs made by this manufacturer differs from 54 months? Perform a two-tailed test. Then fill in the table below. Carry your intermediate computations to at least three decimal places, and round your responses as specified in the table.

The null hypothesis

The alternative hypothesis

The type of test statistic ( choose between z, t, chi squarea and f)

The value of the test statistic:
(Round to at least three decimal places.)

The two critical values at the 0.05 level of significance:
(Round to at least three decimal places.) _________and_________

Can we conclude that the mean lifetime of light bulbs made by this manufacturer differs from 54 months? Yes or No

Answer 1

Here, manufacturer claims that mean lifetime of bulb is 54 months. We have to check that the mean lifetime of bulbs made by this manufacturer differs from 54 months.

So, hypothesis can be defined as -

Null hypothesis - H0 : = 54 i.e. lifetime of bulb does not differ from 54 months.

Alternative hypothesis - H0 :   54 i.e. lifetime of bulb differ from 54 months.

Since, 150 bulbs are selected at random, so we have to use Z-test as sample size is large (n> 30).

Test statistic -

Z =

Test criterion -

The variable Z N(0,1) for the two tailed test reject H0 if Z Z/2 or Z -Z/2.

Calculations -

Here, The mean lifetime of bulbs is found 53 months from selected 150 bulbs. Standard deviation is 8 months.

= 53 months, = 8 months, n = 150

So, Z =

=

Z = -1.5309

Critical value -

We have to test it at 5% level of significance, so -

Z = Z/2 = Z0.05/2 = Z0.025 = 1.96

Two critical values are -1.96 & 1.96.

Conclusion -

Here,Z (-1.5309) > Z/2  (-1.96), so we accept H0 at 5% level of significance. i.e. There is enough evidence that lifetime of bulb does not differ from 54 months. The manufacturer claim is valid.