Questions: A computer manufacturer estimates that its cheapest screens will last less than 2.8 years. A random sample of 61 of these screens has a mean life of 2.5 years. The population is normally distributed with a population standard deviation of 0.88 years. At a=0.02, can you support the organization's claim using the p-value?

A computer manufacturer estimates that its cheapest screens will last less than 2.8 years. A random sample of 61 of these screens has a mean life of 2.5 years. The population is normally distributed with a population standard deviation of 0.88 years. At a=0.02, can you support the organization's claim using the p-value?

Transcript text: A computer manufacturer estimates that its cheapest screens will last less than 2.8 years. A random sample of 61 of these screens has a mean life of 2.5 years. The population is normally distributed with a population standard deviation of 0.88 years. At $a=0.02$, can you support the organization's claim using the $p$-value?

Solution

Solution Steps

Step 1: Given Information

We are testing the claim that the mean life of the cheapest screens is less than $2.8$ years. The relevant parameters are:

Sample mean ($\bar{x}$): $2.5$ years
Hypothesized population mean ($\mu_0$): $2.8$ years
Population standard deviation ($\sigma$): $0.88$ years
Sample size ($n$): $61$
Significance level ($\alpha$): $0.02$

Step 2: Calculate Standard Error

The standard error ($SE$) is calculated using the formula: \[ SE = \frac{\sigma}{\sqrt{n}} = \frac{0.88}{\sqrt{61}} \approx 0.1127 \]

Step 3: Calculate Test Statistic

The Z-test statistic is calculated using the formula: \[ Z = \frac{\bar{x} - \mu_0}{SE} = \frac{2.5 - 2.8}{0.1127} \approx -2.6626 \]

Step 4: Calculate P-value

For a left-tailed test, the p-value is determined as: \[ P = T(z) \approx 0.0039 \]

Step 5: Decision Rule

We compare the p-value to the significance level:

If $P < \alpha$, we reject the null hypothesis.
If $P \geq \alpha$, we fail to reject the null hypothesis.

In this case: \[ 0.0039 < 0.02 \] Thus, we reject the null hypothesis.

Step 6: Conclusion

Since we reject the null hypothesis, we support the claim that the mean life of the cheapest screens is less than $2.8$ years.

Final Answer

The claim can be supported. The p-value is $0.0039$, which is less than $\alpha = 0.02$. Therefore, we reject the null hypothesis, and the claim is the alternative hypothesis and is supported.

$\boxed{\text{Yes, the claim can be supported.}}$

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