High School

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State the null and alternative hypotheses for the test.

- Null Hypothesis ([tex]H_0[/tex]): The mean MPG of the van is equal to the manufacturer's rating, [tex]\mu = 36.9[/tex] MPG.
- Alternative Hypothesis ([tex]H_a[/tex]): The mean MPG of the van is less than the manufacturer's rating, [tex]\mu < 36.9[/tex] MPG.

Answer :

To determine whether there is sufficient evidence to support the testing firm's claim that the actual MPG of the vans is less than the manufacturer's rating, we begin by establishing the null and alternative hypotheses.

Step 1: State the Hypotheses

Null Hypothesis (H₀): The mean miles per gallon (MPG) of the vans is equal to the manufacturer's rating.

  • Mathematically, this can be stated as:
    [tex]H_0: \mu = 36.9[/tex]

Alternative Hypothesis (H₁): The mean miles per gallon (MPG) of the vans is less than the manufacturer's rating. This supports the testing firm’s belief.

  • Mathematically, this can be stated as:
    [tex]H_1: \mu < 36.9[/tex]

Explanation

  • Who: The hypotheses are set by an independent testing firm to verify or contradict the manufacturer's claim regarding the van's MPG.

  • What: We are conducting a hypothesis test to evaluate the mean MPG of vans.

  • Why: The testing firm believes the vans perform under the specified 36.9 MPG rating.

  • How: By testing a sample of 260 vans and calculating their mean MPG, statistical analysis can determine if the observed data provides significant evidence against the manufacturer's claim.

Conclusion

By establishing these hypotheses, the testing firm will perform a statistical test to assess the evidence. If the results show that the data significantly support the alternative hypothesis at the 0.05 significance level, the testing firm can claim there is sufficient evidence to suggest the actual MPG is less than the manufacturer's rating.

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