Final answer:
To find the probability that a single randomly selected value is less than 70.1 in a normal distribution with a mean of 70.4 and a standard deviation of 13.3, we standardize the value by subtracting the mean and dividing by the standard deviation. The probability can be found using the standard normal distribution table or calculator. For a randomly selected sample of size 237, the probability that the sample mean is less than 70.1 is found using the central limit theorem and standardizing the value.
Step-by-step explanation:
To find the probability that a single randomly selected value is less than 70.1 in a normal distribution with a mean of 70.4 and a standard deviation of 13.3, we can use the standard normal distribution table or a calculator. First, we need to standardize the value by subtracting the mean and dividing by the standard deviation:
Z = (70.1 - 70.4) / 13.3 = -0.0226
Next, we can look up the probability associated with the z-score -0.0226 in the standard normal distribution table or use a calculator. The probability can also be found using the cumulative distribution function (CDF) of the standard normal distribution:
P(X < 70.1) = 0.4895
To find the probability that a randomly selected sample of size n=237 has a mean less than 70.1, we can use the central limit theorem. The mean of the sample will also have a normal distribution with the same mean as the population, but the standard deviation will be the population standard deviation divided by the square root of the sample size:
Standard deviation (sample mean) = 13.3 / sqrt(237) = 0.8622
Now we can standardize the value 70.1 by subtracting the mean and dividing by the standard deviation:
Z = (70.1 - 70.4) / 0.8622 = -0.3482
We can then look up the probability associated with the z-score -0.3482 in the standard normal distribution table or use a calculator. The probability can also be found using the cumulative distribution function (CDF) of the standard normal distribution:
P(M < 70.1) = 0.3638