1. 15.3 Normal Distribution to Solve For Probabilities By Haley Wehner

2. Vocabulary μ (Greek letter mu): symbol used for the mean of a population x̄ (x-bar): symbol used for the mean of the sample σ: Standard Deviation Sampling Distribution (of a statistic) – the distribution of values taken by the statistic in all possible samples of the same size from the same population Probability Distribution: replacing the relative frequency with probability

3. Formulas You’ll Use in 15.3: The Z-score formula: z = x- x̄ σ NormalCdf (lower limit, upper limit, x̄, σ) Given %: Inverse normal (Invnormal) % -> give Z-Score % x̄, σ -> Invnormal (%, x̄, σ) -> X

4. Example 1: Example 1: The time it takes Ms. White to solve a math problem every day is normally distributed with a mean of 15 seconds and a standard deviation of 5 seconds. Estimate the number of days when she takes A) longer than 20 seconds normalcdf (lower limit, upper limit, x̄, σ) = about 30 days normal cdf (20, 900, 15, 5) = 0.1587 x 190 = about 30 days B) less than 8 seconds about 15 days normalcdf (0, 8, 15, 5) = 0.0794 x 190 = about 15 days C) between 7 and 17 seconds about 114 days normalcdf (7, 17, 15, 5) = 0.6006 x 190 = about 114 days ***Remember that there are 190 school days at Cactus Shadows*** *^^For this problem, you can choose any number higher than 20 for the upper limit. I chose 900.

5. Example 2: Example 2: A company makes containers of salsa each weighing 600g. Their machines fill the containers with weights that are normally distributed with a standard deviation of 6.2g. A bag that contains less than 600g is considered underweight. A) If the company decides to set their machines to fill the containers with a mean of 612g, what fraction will be underweight? A) This problem tells you that the x̄, or mean, equals 612g. The σ is also given, which equals 6.2g. Since they tell you that a 600g container is underweight, you know that the upper limit for this problem is 600g because that is the weight they want each container to be. The lower limit is 0 because any container weighing between 0 and 599 isn’t usable. So, if you plug these numbers into the normalcdf formula, you get: normalcdf (0, 600, 612, 6.2). 0.026.Next, all you have to do is put normalcdf (0, 600, 612, 6.2) into your calculator, and then you get the answer of how many containers will be underweight to be 0.026.

6. Example 2: Example 2: A company makes containers of salsa each weighing 600g. Their machines fill the containers with weights that are normally distributed with a standard deviation of 6.2g. A bag that contains less than 600g is considered underweight. B) If the company wishes the percentage of underweight containers to be at most 5%, what mean setting must they have? Since this problem is asking you to find the mean, you know you can’t use the normalcdf formula, so you instead use the z-score formula. z = x- x̄ σ -1.64= 600- x̄ 6.2 -10.169=600- x̄ -610.168 =- x̄ 610.168 = x̄ is your answer. 610.168 = x̄ is your answer. *To get z in the z-score formula, go to the invnormal section in your calculator, put in the percent in the area section (in this case, 0.05), 1 for σ, and 0 for μ.

7. Example 2: Example 2: A company makes containers of salsa each weighing 600g. Their machines fill the containers with weights that are normally distributed with a standard deviation of 6.2g. A bag that contains less than 600g is considered underweight. C) If they don’t want to set the mean as high as 612, but instead at 609, what standard deviation gives them at most 5% underweight containers? This problem again requires the Z-score formula, but this time you need to solve for the standard deviation instead of the mean, since you’re given 609 to equal the mean. z = x- x̄ σ -1.64= 600-609 σ -1.64σ=-9 σ=5.49 is your answer.

8. Example 3 Using the Empirical Rule: Example 3 Using the Empirical Rule: The time it takes Kim Kardashian to do her hair and makeup per week is normally distributed with a mean of 45 hours and a standard deviation of 2.3 hours. The probability that Kim takes somewhere between 43 and 48 hours is represented by the shaded area in the following diagram. This diagram represents the standard normal curve. A) Write down the values of a and b. a=43-45 2.3 a= -0.87 a= -0.87 b= 48-45 2.3 b= 1.3 a 0 b x y

9. Example 3 Using the Empirical Rule: Example 3 Using the Empirical Rule: The time it takes Kim Kardashian to do her hair and makeup per week is normally distributed with a mean of 45 hours and a standard deviation of 2.3 hours. The probability that Kim takes somewhere between 43 and 48 hours is represented by the shaded area in the following diagram. This diagram represents the standard normal curve. B) Find the probability that Kim takes –i) more than 43 hours 0.81 Normalcdf (43, 600, 45, 2.3) = 0.81 –ii) between 43 and 48 hours 0.71 Normalcdf (43, 48, 45, 2.3) = 0.71 a 0 b x y *^^For this problem, you can choose any number higher than 43 for the upper limit. I chose 600.

10. Example 3 Using the Empirical Rule: Example 3 Using the Empirical Rule: The time it takes Kim Kardashian to do her hair and makeup per week is normally distributed with a mean of 45 hours and a standard deviation of 2.3 hours. The probability that Kim takes somewhere between 43 and 48 hours is represented by the shaded area in the following diagram. This diagram represents the standard normal curve. 90% of the time Kim Kardashian takes t hours. C) i) Represent this information on a standard normal curve diagram, similar to the one shown, indicating clearly the area representing 90%. C) ii) Find the value of t. For this problem, you use the Z-score formula. You’re given 90% in C) i), so you can put in 0.09 in “area” in the invnormal section of your calculator. Then, 1 for σ, and 0 for μ. Z should equal -1.34. t=48.1 hoursz = t- x̄ -1.34 = t- 45 -3.1 = t-45 t=48.1 hours σ 2.3