Question

Assume that a simple random sample has been selected from a normally distributed population and test the given claim. Identify the null and alternative​ hypotheses, test​ statistic, P-value, and state the final conclusion that addresses the original claim.

A safety administration conducted crash tests of child booster seats for cars. Listed below are results from those​ tests, with the measurements given in hic​ (standard head injury condition​ units). The safety requirement is that the hic measurement should be less than 1000 hic. Use a 0.05 significance level to test the claim that the sample is from a population with a mean less than 1000 hic. Do the results suggest that all of the child booster seats meet the specified​ requirement?

637,643,1179,584,517,550

  

Answer 1

Given: Population mean (µ) = 1000.

Sample size(n) = 6

Now, we need to find the sample mean and sample standard deviation by using the given data.

We will get,
Sample mean = xbar = 685
Sample standard deviation = S = 246.8821581

The claim statement is, "the sample is from a population with mean less than 1000 hic".
That is µ < 1000
Hence, the claim statement goes under the alternative hypothesis.

Hence, the Null hypothesis is, H0: µ >= 1000
And the alternative hypothesis is, H1: µ < 1000

Here, we have the sample standard deviation. So, we need to use the T-test for testing.

Now, let's find the test statistic.

That is, test statistic T = -3.1253

Degrees of freedom = n -1 = 6 - 1 = 5 .

Now, let's find the P-value.
For finding the P-value, we can use a technology like excel/Ti84 calculator or T table.
The following excel command is used to find the P-value.

= T.DIST(Test statistic, Degrees of freedom, 1)
= T.DIST(-3.1253 , 5 , 1)

You will get, P-value = 0.01305

We are given: Level of significance α= 0.05

Following is the decision rule for making the decision.
If, P-value > α, then we fail to reject the null hypothesis.
If, P-value <= α, then we reject the null hypothesis.

α = 0.05
And P-value = 0.01305
That is P-value < α.
Hence, we reject the null hypothesis.
That is, there is sufficient evidence to support the claim that the sample is from a population with mean less than 1000 hic.

Hence, the result suggest that all of the child booster seats meet the specified requirements.