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Properties of Normal Distributions 1- The entire family of normal distribution is differentiated by its mean µ and its standard deviation σ. 2- The highest point on the normal curve is at the mean which is also the median and the mode of the distribution. 3- The mean of the distribution can be any numerical value: negative, zero or positive. 4- The normal distribution is symmetric 5- The standard deviation determines how flat and wide the curve is 6- Probabilities for the random variables are given by areas under the curve. The total area under the curve for the normal distribution is 1 7- Because the distribution is symmetric, the area under the curve to the left of the mean is 0.50 and the area under the curve to the right of the mean is 0.50 8- The percentage of values in some commonly used intervals are; a-) 68.3% of the values of a normal r.v are within plus or minus one st.dev. of its mean b-) 95.4% of the values of a normal r.v are within plus or minus two st.dev. of its mean c-) 99.7% of the values of a normal r.v are within plus or minus three st.dev. of its mean Find the area under the standard normal curve that lies 1- to the right of z = - 0.55 2- to the left of z = 0.84 3- to the right of z = 4- to the left of 1.69 z = - 0.74 5- between z = 0.90 and z= 1.33 6- between z = -0.29 and z= 0.59 A rural enterprise center provides an advice service to small craft businesses on all aspects of the business, such as production methods, marketing accounts etc. A candle making business incurs losses as a result of candle glasses being overfilled or under filled (hence poor quality) when colored candle wax is poured into the glass tubes. The wax is filled to heights that are normally distributed with µ1 = 25 cm and σ1 = 0.5 cm and the heights of the tubes are normally distributed with µ2 = 26 cm and σ2 = 0.4 cm. Calculate the probability that height of the gap between the top of the tube and the wax is between 0.5 cm and 1.5 cm. A sociologist has been studying the criminal justice system in a large city. Among other things, she has found that over the last 5 years the length of time an arrested person must wait between their arrest and their trial is a normally distributed variable x with µ=210 days and σ=20 days. Q-) What percent of these people had their trial between 160 days and 190 days after their arrest? Q-1 - MOBILE PHONE Assume that the length of time, x, between charges of mobile phone is normally distributed with a mean of 10 hours and a standard deviation of 1.5 hours. Find the probability that the mobile phone will last between 8 and 12 hours between charges. Q-2 ALKALINITY LEVEL The alkalinity level of water specimens collected from the River in a country has a mean of 50 milligrams per liter and a standard deviation of 3.2 milligrams per liter. Assume the distribution of alkalinity levels is approximately normal and find the probability that a water specimen collected from the river has an alkalinity level A-) exceeding 45 milligrams per liter. B-) below 55 milligrams per liter. C-) between 51 and 52 milligrams per liter. The grades of 400 students in a statistics course are normally distributed with mean µ = 65 and variance σ2=100 Q: Find the probability that a student selected randomly from this group would score within any interval given below. 1- A grade between 60 and 65 2- A grade between 70 and 65 3- A grade between 52 and 68 4- A grade that is greater than 85 5- A grade that is less than 72 6- A grade between 70 and 78 Example : 1 A normal distribution has the mean 74.4 Find its standard deviation if 10% of the area under the curve lies to the right of 100 Example : 2 A random variable has a normal distribution with standard deviation 10 . Find its mean if the probability is 0.8264 that it will take on a value less than 77.5 Example :3 For a certain random variable having the normal distribution, the probability is 0.33 that it will take on a value less than 245 and the probability is 0.48 that it will take on a value greater than 260. Find the mean and standard deviation of the random variable