Hemoglobin is a protein containing iron that facilitates the transport of oxygen in red blood cells. Hemoglobin levels in females are normally distributed and low levels are often associated with incidence of anemia. To assess anemia prevalence, a study is done to test if hemoglobin levels in female residents of a rural region are significantly different from the national average of \( \mu=13.2 \) grams/deciliter. The hemoglobin levels of a simple random sample of 17 female residents were collected and analyzed. Are the requirements for finding a confidence interval or conducting a hypothesis test for the hemoglobin levels study met? No, the sample values were randomly selected, but the distribution of sample means is not normal because the sample size is not large enough. No. the distribution of sample means is normally distributed, but the sample values were not randomly selected. Yes, the sample values were randomly selected, and the distribution of sample means is normal because the population is normally distributed. Yes, the sample values were randomly selected, and the distribution of sample means is normal because the sample size is large enough.
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When it comes to the statistical requirements for conducting hypothesis tests or finding confidence intervals, one crucial principle is the Central Limit Theorem. If the population from which your sample is drawn is normally distributed, the sample means will also be normally distributed, regardless of sample size. However, if the population distribution is unknown, a sample size of 30 or greater is typically recommended to ensure that the distribution of the sample means will approximate normality. In this scenario, since there is a small sample of only 17 female residents, the sample means' distribution may not be normally distributed unless the underlying population of hemoglobin levels is itself relatively normal. Therefore, if it is known that the population is reasonably normally distributed, the sample might meet the necessary conditions for the hypothesis test. Otherwise, while random selection is important, the sample size is a key factor in determining the suitability of the analysis.
