Lightbulbs of a certain type are advertised as having an...

Lightbulbs of a certain type are advertised as having an average lifetime of 750 h. The price of these bulbs is very favorable, so a potential customer has decided to go ahead with a purchase arrangement unless it can be conclusively demonstrated that the average lifetime is smaller than what is advertised. A random sample of 50 bulbs was selected and the lifetime of each bulb determined. These 50 light bulbs had a sample mean lifetime of 738.44 h with a sample standard deviation of 38.20 h. What conclusion would be appropriate for a significance level of .05$?$

Lightbulbs of a certain type are advertised as having an average lifetime of 750 h. The price of
these bulbs is very favorable, so a potential customer has decided to go ahead with a purchase
arrangement unless it can be conclusively demonstrated that the average lifetime is smaller
than what is advertised. A random sample of 50 bulbs was selected and the lifetime of each bulb
determined. These 50 light bulbs had a sample mean lifetime of 738.44 h with a sample standard
deviation of 38.20 h. What conclusion would be appropriate for a significance level of .05$?$

Key Concepts

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population parameter based on sample data. It involves setting up a null hypothesis that represents a default or status quo claim and an alternative hypothesis that represents a competing claim. Statistical tests are then used to determine whether the observed sample data are sufficiently inconsistent with the null hypothesis to justify rejecting it.

Null and Alternative Hypotheses

The null hypothesis is a statement that there is no effect or no difference, often representing a baseline claim, while the alternative hypothesis represents a deviation from this baseline. In testing claims about means, the null hypothesis might state that the true mean is equal to a specific advertised value, and the alternative hypothesis might state that the true mean is lower or higher than that value, depending on the claim being tested.

One-Sample t-Test

A one-sample t-test is used to compare the mean of a single sample to a known value, particularly when the population standard deviation is unknown and the sample size is relatively small. It calculates a t-statistic that quantifies how far the sample mean deviates from the hypothesized population mean in terms of estimated standard error.

Significance Level

The significance level, often denoted by alpha (?), is the threshold for determining whether the test result is statistically significant. It represents the maximum probability of making a Type I error, which is rejecting the null hypothesis when it is actually true. A significance level of 0.05 indicates that there is a 5% risk of incorrectly rejecting the null hypothesis.

Test Statistic and p-Value

The test statistic, in this case a t-statistic, is computed by dividing the difference between the sample mean and the hypothesized population mean by the standard error of the sample mean. The resulting p-value indicates the probability of obtaining a test statistic at least as extreme as the one observed, assuming that the null hypothesis is true. A p-value less than the significance level suggests that the observed result is unlikely under the null hypothesis, leading to its rejection.

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