Answer to Question #259224 in Statistics and Probability for Lala

Question #259224

The manufacturer of television tubes knows from the past experience that the average life of tube is 2000 hrs with a s.d. of 200 hrs. A sample of 100 tubes has an average life of 1950 hrs. Test at the 0.01 level of significance to see if this sample came from a normal population of mean 2000 hrs.

1
Expert's answer
2021-11-01T16:52:02-0400

In this question, we determine whether the population mean"(\\mu)" is equal to 2000hrs as claimed. To do so, we shall test the following hypotheses,

"H_0:\\mu=2000"

"Against"

"H_1:\\mu\\not=2000"

We are given the following information,

"n=100"

"\\bar{x}=1950"

"\\sigma=200" and "\\alpha=0.01"

To perform this test, we shall use the standard Normal distribution since our sample size is large. We proceed as follows.

The test statistic is given as,

"Z^*=(\\bar{x}-\\mu)\/(\\sigma\/\\sqrt{n})"

"Z^*=(1950-2000)\/(200\/\\sqrt{100})=-50\/20=-2.5"

The test statistic "Z^*" is compare with the table value at "\\alpha=0.01" significance level. This table value is given as,

"Z_{\\alpha\/2}=Z_{0.01\/2}=Z_{0.005}=2.575" and the null hypothesis is rejected if "|Z^*|\\gt Z_{0.005}"

Since "|Z^*|=|-2.5|=2.5\\lt Z_{0.005}=2.575", we fail to reject the null hypothesis and we conclude that there is sufficient evidence to show that this sample came from a normal population of mean 2000 hours at 1% level of significance.


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