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A battery manufacturer claims their batteries last an average of 100 hours. A sample of 60 batteries lasts an average of 98.5 hours with σ = 4. Test the claim.

Answer :

To test the battery manufacturer's claim that their batteries last an average of 100 hours, we can perform a hypothesis test. Here's the step-by-step process:

  1. State the Hypotheses:

    • Null hypothesis [tex]H_0[/tex]: The average battery life is 100 hours, [tex]\mu = 100[/tex].
    • Alternative hypothesis [tex]H_a[/tex]: The average battery life is not 100 hours, [tex]\mu \neq 100[/tex]. This is a two-tailed test because we are checking for any deviation from 100 hours.

  2. Set the Significance Level:

    • Commonly, a significance level ([tex]\alpha[/tex]) of 0.05 is used. We'll use this unless another level is specified.

  3. Calculate the Test Statistic:
    Use the formula for the z-test statistic for a sample mean:
    [tex]z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}[/tex]
    Where:

    • [tex]\bar{x} = 98.5[/tex] is the sample mean.
    • [tex]\mu = 100[/tex] is the claimed population mean.
    • [tex]\sigma = 4[/tex] is the population standard deviation.
    • [tex]n = 60[/tex] is the sample size.

    Plugging in the values:
    [tex]z = \frac{98.5 - 100}{\frac{4}{\sqrt{60}}} = \frac{-1.5}{0.5164} \approx -2.90[/tex]

  4. Find the Critical Value:

    • For a two-tailed test at [tex]\alpha = 0.05[/tex], we check the standard normal distribution table (Z-table) for critical z-values. The critical values are approximately [tex]\pm 1.96[/tex].

  5. Make a Decision:

    • Compare the calculated z-value of approximately [tex]-2.90[/tex] with the critical values. Since [tex]-2.90[/tex] is less than [tex]-1.96[/tex], it falls in the critical region.

  6. Conclusion:

    • We reject the null hypothesis [tex]H_0[/tex]. There is sufficient evidence to suggest that the average battery life is different from 100 hours.

Therefore, based on this test, the manufacturer's claim that the batteries last, on average, 100 hours is not supported by the sample data.

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