Final answer:
To find the values of x_ _ x and x_ , you can use the formulas x_ _ x = + (z * τ) and x_ = + (z * (τ/√n)). To calculate the standard normal z-scores corresponding to x = 68 and x = 61, use the formula z = (x - ) / τ. To find the probabilities P(>67) and P(62<x<68), convert the values to z-scores and use a standard normal distribution table or calculator.
Step-by-step explanation:
To find the value of x_ _ x, we can use the formula: x_ _ x = + (z * τ), where is the population mean, z is the z-score, and τ is the standard deviation. Given that the population mean is 65 and the standard deviation is 12, we can substitute these values into the formula. Similarly, to find the value of x_ , we can use the formula: x_ = + (z * (τ/√n)), where n is the sample size. Given that the sample size is 64, we can substitute this value into the formula.
To calculate the standard normal z-score corresponding to a value of x = 68, we can use the formula: z = (x - ) / τ. Given that x = 68 and the population mean is 65, we can substitute these values into the formula. Similarly, to calculate the standard normal z-score corresponding to x = 61, we can use the same formula and substitute x = 61 and the population mean, , which is 65.
To find P(>67), we need to convert the value to a z-score and then use a standard normal distribution table or calculator to find the corresponding probability. The formula to calculate the z-score is the same as mentioned earlier: z = (x - ) / τ. Given that x = 67 and the population mean is 65, we can substitute these values into the formula. Similarly, to find P(62<x<68), we need to calculate the individual probabilities and then subtract the lower probability from the higher probability. We can convert the values to z-scores using the same formula and then use a standard normal distribution table or calculator to find the corresponding probabilities.