normal probability distribution
sampling distribution of a statistic
(such as a sample mean or sample proportion) is the distribution of all values of the statistic when all possible samples of the same size n are taken from the same population. The sampling distribution of a statistic is typically represented as a probability distribution in the format of a table, probability histogram, or formula.
A recent botany experiment showed a variety of results for the height of certain plants. The results followed a normal distribution with a mean of 8.4 feet and a standard deviation of 2.2 feet. Find the probability that a randomly selected plant with have a height less than 6.0 feet.
0.1379
A type of thermometer has temperature readings at the freezing point of water that are normally distributed with a mean of 0°C and a standard deviation of 1.00°C. Find the temperature corresponding to P68 upper P subscript 68 baseline, the 68th percentile.
0.47
Find the probability of selecting a z score less than 0.12.
0.5478
After recording the maximum distance possible when driving a new electric car, the study showed the distances followed a normal distribution. The mean distance is 134 miles and the standard deviation is 4.8 miles. Find the probability that in a random test run the car will travel a maximum distance between 125 and 135 miles.
0.5531
A type of thermometer has temperature readings at the freezing point of water that are normally distributed with a mean of 0°C and a standard deviation of 1.00°C. If a thermometer is randomly selected, find the probability that at the freezing point of water, the selected thermometer reads above -1.96° negative 1.96 degrees.
0.9750 area left to the z score -1.96 is 0.0250 1 - 0.0250 = 0.9750
Assuming a standard normal distribution, find the area between -2.87 and 1.34.
Find the area less than 1.34 and the area less than -2.87. Area = 0.9099 - 0.0021 = 0.9078
z scores and area
Remember, z scores are distances along the horizontal scale, whereas areas (or probabilities) are regions under the curve. In Table A-2, z scores are listed in the left column and across the top row, but areas are found in the body of the table. Also, z scores positioned in the left half of the curve are always negative.
procedure for finding values
Sketch a normal distribution curve, enter the given probability or percentage in the appropriate region of the graph, and identify the x value(s) being sought. Use Table A-2 to find the z score corresponding to the cumulative left area bounded by x. Refer to the body of Table A-2 to find the closest area, then identify the corresponding z score. Using the conversion formula, enter the values for μ, σ, and the z score found in Step 2, then solve for x. We can solve for x as follows: x = μ + (z • σ)x equals mu plus left parenthesis z times sigma right parenthesis (another form of the conversion formula) Refer to the sketch of the curve to verify that the solution makes sense in the context of the graph and in the context of the problem.
standard normal distribution
normal probability distribution that has a mean of 0 and a standard deviation of 1, and the total area under its density curve is 1.
uniform distribution
probability distribution in which every value of the random variable is equally likely
estimators
sample statistic (such as the sample mean x⎯⎯ , used to approximate a population parameter) some statistics (such as the mean, variance, and proportion) are unbiased estimators of population parameters, whereas other statistics (such as the median and range) are estimators.
A factory has a machine that makes steel rods. The length of the rods that the machine makes follows a normal distribution curve with a mean of 11.5 feet and a standard deviation of 0.3 feet. The factory manager does not want to use the rods from the bottom 20% and the top 15%. Find the rod lengths the manager will be using as the basis for separating the rods.
11.25 and 11.81
A girl scout troop sold cookies to raise funds. Each troop member recorded the number of boxes she sold. The number of boxes sold was normally distributed with a mean of 22 boxes sold and a standard deviation of 4 boxes. In order to help determine how many boxes would be sold the next year, the troop leader decided to focus on the range made from the bottom 30% to the top 20%. Find the number of boxes that represent the desired range values.
20 and 25
Several science students used the same design when constructing a bridge out of balsa wood. They each tested the bridge's weight capacity using cups filled with sand. The weight capacities were recorded, and the results follow a normal distribution curve. The mean of the weights is 19.6 lb with a standard deviation of 1.3 lb. Which weight is greater than 70% of the data?
20.29 lb
uniform distributions
A continuous random variable has a uniform distribution if its values are spread evenly over the range of possibilities. The graph of a uniform distribution results in a rectangular shape. The uniform distribution allows us to see two very important properties: The area under the graph of a probability distribution is equal to 1. There is a correspondence between area and probability (or relative frequency), so some probabilities can be found by identifying the corresponding areas.
Which is not a property of the standard normal distribution?
All data values fall within 3 standard deviations of the mean In a standard normal distribution, about 99.7%, but not all, data values fall within 3 standard deviations of the mean.
density curve (probability density function)
a graph of a continuous probability function. the total area under the density curve must equal to 1, every point on the curve must have a height of 0 or greater.
critical value
a z score on the borderline separating the z scores that are likely to occur from those that are unlikely. The values below z=−1.96 z equals negative 1.96 are not likely to occur, because they occur in only 2.5% of the readings, and the values above z = 1.96 are not likely to occur because they also occur in only 2.5% of the readings.
