If 12 women are randomly selected, what is the probability that their mean head circumference is between 21.8 in and 22.8 in.? If this probability is high does it suggest that an order of 12 hats will very likely for each of 12 randomly selected women? Why or why not? Assume that they fit head circumferences between 21.8 and 22. 8 in
To calculate the probability that the mean head circumference of 12 randomly selected women falls between 21.8 in and 22.8 in, we need to know the population parameters such as the mean and standard deviation of head circumferences. Let's assume the mean head circumference is represented by μ and the standard deviation by σ.
Next, we need to rely on a statistical concept called the Central Limit Theorem, which states that the distribution of sample means approximates a normal distribution, regardless of the shape of the population distribution, as long as the sample size is sufficiently large (typically above 30). In this case, we have a sample size of 12, which is smaller than the commonly recommended sample size.
If the population distribution is normal or close to normal, and the sample size is large enough, we can make an approximation using the z-score formula:
z = (x - μ) / (σ / √n)
where x is the desired mean value, μ is the population mean, σ is the population standard deviation, and n is the sample size.
In this case, since we don't have the population parameters (μ and σ), it's not possible to calculate the exact probability. However, we can estimate it if we assume a normal distribution for head circumferences.
Now let's address the second part of your question. The probability of the mean head circumference falling between 21.8 in and 22.8 in does not directly suggest that an order of 12 hats will very likely fit each of the 12 randomly selected women. This is because the mean head circumference only represents the average, and individual head circumferences can still vary widely around that mean.
To determine the likelihood of hats fitting each woman, we need more information about the distribution of head circumferences, such as the range, variability, and distribution shape. Additionally, using a sample size of 12 may not be representative enough to make accurate predictions for individual women.
In summary, the probability of the mean head circumference falling between 21.8 in and 22.8 in does not directly relate to the hats fitting each individual woman. To determine the likelihood of hat fitting, we would need more specific information about the head circumference distribution and a larger sample size.