A company manufactures light bulbs. The company wants the...

A company manufactures light bulbs. The company wants the bulbs to have a mean life span of 1000 hours. This average is maintained by periodically testing random samples of 16 light bulbs. If the $t$ -value falls between $-t_{0.99}$ and $t_{0.99},$ then the company will be satisfied that it is manufacturing acceptable light bulbs. A sample of 16 light bulbs is randomly selected and tested. The mean life span of the sample is 1015 hours and the standard deviation is 25 hours. Assume the life spans are approximately normally distributed. Is the company making acceptable light bulbs? Explain your reasoning.

Key Concepts

Degrees of Freedom

Degrees of freedom refer to the number of independent values in a calculation that are free to vary while estimating a statistical parameter. In the context of the t-distribution, the degrees of freedom are typically equal to the sample size minus one. This parameter influences the shape of the t-distribution, with fewer degrees of freedom leading to a distribution that is more spread out.

Hypothesis Testing

Hypothesis testing is a procedure used to determine whether sample data provide sufficient evidence to support or refute a claim about a population parameter. It involves stating a null hypothesis and an alternative hypothesis, then using a test statistic calculated from sample data to decide if the observed result is statistically significant when compared to a critical value from the relevant distribution.

t-Distribution

The t-distribution is used when dealing with small sample sizes or when the population standard deviation is unknown. It resembles the normal distribution but has thicker tails, which account for the extra variability expected from estimating the population standard deviation from a sample. This distribution is essential in the context of the one-sample t-test, where sample statistics are used to make inferences about the population mean.

Critical Values and Rejection Regions

Critical values define the threshold boundaries of a test statistic beyond which the null hypothesis is rejected. In a two-tailed test, these values create an acceptance region within which the computed test statistic is considered consistent with the null hypothesis. If the test statistic falls outside this region, it provides evidence against the null hypothesis.

Standard Error

The standard error measures the dispersion of sample statistics around the true population parameter, quantifying the variability of the sample mean from sample to sample. It is computed as the sample standard deviation divided by the square root of the sample size, and it is crucial for constructing confidence intervals and calculating the test statistic in hypothesis testing.

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