Engineers must consider the breadths of male heads when designing helmets. The company researchers have determined that the population of potential clientele have head breadths that are normally distributed with a mean of 6.5 -in and a standard deviation of 0.9 -in. In what range would you expect to find the middle \( 68 \% \) of most head breadths? Between \( \square \) and If you were to draw samples of size 40 from this population, in what range would you expect to find the middle \( 68 \% \) of most averages for the breadths of male heads in the sample? Between \( \square \) and Enter your answers as numbers. Your answers should be accurate to 2 decimal places.

To find the range for the middle 68% of most head breadths, we can use the empirical rule, which states that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean. Given: - Mean (μ) = 6.5 in - Standard Deviation (σ) = 0.9 in For the population: - Lower limit: μ - σ = 6.5 - 0.9 = 5.6 in - Upper limit: μ + σ = 6.5 + 0.9 = 7.4 in So, you would expect to find the middle 68% of most head breadths between **5.6** and **7.4** inches. Now, when drawing samples of size 40, we need to consider the standard error (SE) of the sample mean, which is calculated using the formula SE = σ / √n. Given: - Sample size (n) = 40 - σ = 0.9 in Calculating the SE: SE = 0.9 / √40 ≈ 0.142 Now, we can find the range for the middle 68% of the sample means: - Lower limit: μ - SE = 6.5 - 0.142 = 6.358 - Upper limit: μ + SE = 6.5 + 0.142 = 6.642 You would expect to find the middle 68% of most averages for the breadths of male heads in the sample between **6.36** and **6.64** inches.