(a) identify the claim and state H0 and Ha,(b) find the...

(a) identify the claim and state $H_{0}$ and $H_{a},(b)$ find the critical value(s) and identify the rejection region$(s),(c)$ find the standardized test statistic $z,(d)$ decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. If convenient, use technology. A fluorescent lamp manufacturer guarantees that the mean life of a certain type of lamp is at least 10,000 hours. You want to test this guarantee. To do so, you record the lives of a random sample of 32 fluorescent lamps. The results (in hours) are listed. Assume the population standard deviation is 1850 hours. At $\alpha=0.09,$ do you have enough evidence to reject the manufacturer's claim? $$\begin{array}{rlrlrrll} 8,800 & 9,155 & 13,001 & 10,250 & 10,002 & 11,413 & 8,234 & 10,402 \\ 10,016 & 8,015 & 6,110 & 11,005 & 11,555 & 9,254 & 6,991 & 12,006 \\ 10,420 & 8,302 & 8,151 & 10,980 & 10,186 & 10,003 & 8,814 & 11,445 \\ 6,277 & 8,632 & 7,265 & 10,584 & 9,397 & 11,987 & 7,556 & 10,380 \end{array}$$

(a) identify the claim and state $H_{0}$ and $H_{a},(b)$ find the critical value(s) and identify the rejection region$(s),(c)$ find the standardized test statistic $z,(d)$ decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. If convenient, use technology.
A fluorescent lamp manufacturer guarantees that the mean life of a certain type of lamp is at least 10,000 hours. You want to test this guarantee. To do so, you record the lives of a random sample of 32 fluorescent lamps. The results (in hours) are listed. Assume the population standard deviation is 1850 hours. At $\alpha=0.09,$ do you have enough evidence to reject the manufacturer's claim?
$$\begin{array}{rlrlrrll}
8,800 & 9,155 & 13,001 & 10,250 & 10,002 & 11,413 & 8,234 & 10,402 \\
10,016 & 8,015 & 6,110 & 11,005 & 11,555 & 9,254 & 6,991 & 12,006 \\
10,420 & 8,302 & 8,151 & 10,980 & 10,186 & 10,003 & 8,814 & 11,445 \\
6,277 & 8,632 & 7,265 & 10,584 & 9,397 & 11,987 & 7,556 & 10,380
\end{array}$$

Key Concepts

Hypotheses Formulation

This concept involves setting up a null hypothesis (H0) that represents the default or status quo condition and an alternative hypothesis (Ha) that represents the claim being tested. The formulation of these hypotheses is crucial as they define the framework for assessing whether observed data provide enough evidence to support a departure from the status quo.

One-Sample Hypothesis Testing

This is a statistical procedure used to determine whether the mean or proportion of a single population differs from a known value. In this method, data from a random sample are used to make inferences about the population parameter, and the process usually assumes conditions such as normality or a large sample size to justify the use of parametric tests.

Standardized Test Statistic

The standardized test statistic, often computed as a z-score when the population standard deviation is known, converts the sample mean into a value that can be compared with a standard normal distribution. This transformation allows statisticians to quantify how far the observed statistic is from the null hypothesis in units of standard error.

Critical Value and Rejection Region

Critical values are threshold values determined by the significance level (alpha) which define regions where the null hypothesis would be rejected. The rejection region is the range of values for which, if the computed test statistic falls within, the null hypothesis is rejected, indicating significant evidence against it.

Decision Rule and Interpretation

The decision rule in hypothesis testing specifies that if the test statistic falls into the rejection region defined by the critical value(s), the null hypothesis is rejected; otherwise, it is not rejected. Interpreting this decision requires considering the context of the test—understanding that rejecting H0 suggests that the observed data are unlikely under the assumption of the null hypothesis, while failing to reject H0 indicates insufficient evidence to support the alternative.

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