Therefore, the probability of the temperature is between 70 and 100 degrees Fahrenheit is 0.8664.
To find these probabilities
The conventional normal distribution (Z) will be employed. With a mean of 0 and a standard deviation of 1, the standard normal distribution is a Gaussian distribution. Any Gaussian random variable X can be made standard by deducting its mean (µ) and dividing the result by its standard deviation (σ):
Z = (X - µ) / σ
The probability can be found using a calculator or the standard normal distribution table once the random variable has been normalized.
P[T > 100]
First, we standardize the value 100:
Z = (100 - 85) / 10 = 1.5
P(Z > 1.5) ≈ 0.9332, according to the conventional normal distribution table. Consequently, there is a about 0.9332 chance that the temperature is higher than 100 degrees Fahrenheit.
P[T < 60]
First, we standardize the value 60:
Z = (60 - 85) / 10 = -2.5
P(Z < -2.5) = 0.0062, according to the conventional normal distribution table. Consequently, there is a 0.0062 chance that the temperature is below 60 degrees Fahrenheit.
P[70 < T < 100]
By deducting the two probabilities we previously computed, we may determine this probability:
P[70 < T < 100] = P(T < 100) - P(T < 70)
P(T < 100) = 0.9332 (calculated above)
To find P(T < 70), we standardize the value 70:
Z = (70 - 85) / 10 = -1.5
P(Z < -1.5) = 0.0668, according to the conventional normal distribution table. P(T < 70) ≈ 0.0668 as a result.
P[70 < T < 100] = 0.9332 - 0.0668 ≈ 0.8664