To solve this question, we first need to calculate the z-score that represents the number of standard deviations a data point is from the mean. For this, we use the following formula:
Z = (X - μ) / (σ/√n),
where X is the sample mean, μ is the population mean, σ is the standard deviation, and n is the sample size. Substituting in the known values, we have:
Z = (990 - 1000) / (20/√26)
This calculation gives us a z-score of -2.5495097567963922.
The negative z-score indicates that the sample mean is less than the population mean. A z-score of -2.55 roughly means that the sample mean (990 hours) is around 2.55 standard deviations below the population mean (1000 hours).
Next, we calculate the probability (p-value) of getting a sample mean of 990 hours (or less) under the null hypothesis that the population mean is 1000 hours. This is the area under the bell curve to the left of the z-score. We use a standard normal distribution table or calculator to find the probability associated with our calculated z-score.
For our z-score, the corresponding p-value is 0.005393724627335183.
This gives us a result of about 0.0054 or 0.54%.
This p-value is less than 0.05 (or the significance level in many statistical tests), which suggests that the observed difference between the mean of the sample and the population is statistically significant. Therefore, originating from a population with a mean of 1000 hours, we would expect to get a sample mean of 990 hours (or less) just around 0.54% of the time, assuming that the variability in the population is the same as in our sample.
So in conclusion, given the small p-value, there is strong evidence against the null hypothesis that the population mean is 1000 hours. We would say that the light bulbs' life span in the sample is significantly different (shorter) than that claimed by the manufacturer. Hence, the sample is not up to standard.