The x distribution is a normal distribution with a mean of 70 and a standard deviation of 9. The z value corresponding to x = 67 is -0.33. The probability of x being less than 67 is approximately 0.3707.
Step-by-step explanation:
The x distribution is a normal distribution with a mean of 70 and a standard deviation of 9.
To find the z value corresponding to x = 67, we subtract the mean from x and divide by the standard deviation. So, (67-70)/9 = -0.33. Therefore, z = -0.33.
To find P(x < 67), we need to find the z score corresponding to x = 67 and then find the area under the curve to the left of that z score. Using a z-table or calculator, we find that P(z < -0.33) ≈ 0.3707.
Since the distribution of sample means follows a normal distribution with the same mean as the population mean but a smaller standard deviation (standard deviation of the population divided by the square root of the sample size), it is unlikely for a random sample of size 36 to have a sample mean less than 67. Therefore, it would be unusual for a random sample of size 36 to have a sample mean less than 67.