7.3: The Normal Probability Distribution

Question 1

What is the normal probability distribution?

Answer 1

The normal probability distribution, also known as the normal model, is a bell-shaped curve that is symmetrical around the mean μ and describes how continuous data are distributed, where most values cluster around a central region and the probabilities for values further away from the mean taper off symmetrically in both directions.

Question 2

What is the equation that defines the normal probability distribution?

Answer 2

f(x) = (1 / (σ√2π)) * e^(-1/2 * ((x - μ) / σ)^2) for all values of x on the real line, where μ is the mean, σ is the standard deviation, π is approximately 3.14159, and e is the base of the natural logarithm, approximately 2.71828.

Question 3

What are the key properties of the normal distribution?

Answer 3

1) It has a bell-shaped curve with a peak at the mean μ, which is also the median and mode.

2) It is symmetrical about the mean.

3) The tails of the curve approach the horizontal axis asymptotically.

4) The total area under the curve equals 1.

5) The area under the curve to the left of the mean is equal to 0.5, as is the area to the right of the mean.

Question 4

How does the mean μ affect the position of a normal curve?

Answer 4

The mean μ positions the normal curve along the x-axis; normal curves with different means will be centered at different points along the x-axis.

Question 5

How do the mean μ and standard deviation σ affect the shape of the normal probability curve?

Answer 5

A greater mean μ shifts the curve to the right on the x-axis, while a greater standard deviation σ makes the curve flatter and more spread out.

Question 6

How is the probability of a random variable x being between two values a and b found using the normal curve?

Answer 6

The probability P(a ≤ x ≤ b) is the area under the normal curve between points a and b.

Question 7

What does the Empirical Rule state for a normal distribution?

Answer 7

The Empirical Rule states that approximately 68.26% of the data will lie within ±1 standard deviation from the mean, 95.44% within ±2 standard deviations, and 99.73% within ±3 standard deviations.

Question 8

What are the key properties of the normal distribution?

Answer 8

The normal distribution is defined by the following properties:

1) It has a bell-shaped curve;
2) It is symmetrical around the mean
μ;
3) Its mean, median, and mode are all equal;
4) The tails of the distribution extend to infinity;
5) The total area under the curve equals 1.

Question 9

How does the mean μ and standard deviation σ affect the shape of the normal curve?

Answer 9

The mean μ determines the center of the curve, and the standard deviation σ affects the spread or width of the curve.

Larger σ results in a flatter and wider curve, while a smaller σ leads to a steeper curve.

Question 10

How is the standard normal distribution defined?

Answer 10

The standard normal distribution is a special case of the normal distribution with a mean of 0 and a standard deviation of 1.

It is used to find probabilities and areas under the curve for any normal distribution through the process of standardization.

Question 11

What is the z-score in a normal distribution?

Answer 11

The z-score is the number of standard deviations an observation x is from the mean μ.

It is calculated as

z= (x−μ) /σ

Question 12

How do you find probabilities using the standard normal table?

Answer 12

To find the probability that z is less than or equal to a value, locate the z-score in the standard normal table, which gives the area under the curve to the left of z.

For right-tail probabilities, subtract the table value from 1.

Question 13

What does the cumulative normal table provide?

Answer 13

The cumulative normal table provides the area under the standard normal curve to the left of a given z-score, representing the probability that z is less than or equal to that z-score.

Question 14

How can you find the probability of a z-score between two values using the cumulative normal table?

Answer 14

To find P(a≤z≤b), calculate the area to the left of b (from the table) and subtract the area to the left of a. This gives the area between a and b under the standard normal curve.

Question 15

What is the purpose of the normal table?

Answer 15

The normal table is used to find the area under the standard normal curve, which corresponds to the probability of z-values (standardized values).

Question 16

What is the purpose of the normal table?

Answer 16

The normal table is used to find the area under the standard normal curve, which corresponds to the probability of z-values (standardized values).

Question 17

How do you calculate P(1 ≤ z ≤ 2)?

Answer 17

P(1 ≤ z ≤ 2) is calculated by subtracting the area to the left of z=1 from the area to the left of z=2, using the standard normal table

Question 17

What does P(−∞ < z < ∞) equal in a normal distribution?

Answer 17

P(−∞ < z < ∞) equals 1, as the total area under the normal curve equals 1.

Question 18

What percentage of values lies within plus or minus 1 standard deviation in a normal distribution?

Answer 18

Approximately 68.26% of values lie within plus or minus 1 standard deviation in a normal distribution.

Question 19

What percentage of values lies within plus or minus 2 standard deviations in a normal distribution?

Answer 19

Approximately 95.44% of values lie within plus or minus 2 standard deviations in a normal distribution.

Question 20

What percentage of values lies within plus or minus 3 standard deviations in a normal distribution?

Answer 20

Approximately 99.73% of values lie within plus or minus 3 standard deviations in a normal distribution.

Question 21

How do you find the probability of a z-score between any two values?

Answer 21

To find P(a ≤ z ≤ b), subtract the area to the left of z=a from the area to the left of z=b in the standard normal table.

Question 22

What is the area under the standard normal curve between −1.96 and 1.96?

Answer 22

The area under the standard normal curve between −1.96 and 1.96 is approximately 0.9500, representing 95% of the probability.

Question 23

How do you use the normal distribution to compute probabilities?

Answer 23

To compute probabilities using the normal distribution, convert the problem into terms of the standard normal variable z, then find the required area under the standard normal curve using the normal table.

Question 24

In the context of car mileage, what does a z-score represent?

Answer 24

A z-score represents the number of standard deviations a car’s mileage is from the mean mileage.

Question 25

How is the z-score calculated?

Answer 25

The z-score is calculated with the formula z = (x - μ) / σ, where x is the observed value, μ is the mean, and σ is the standard deviation.

Question 26

How do you find the probability for a range of values using the normal distribution?

Answer 26

To find the probability for a range of values, calculate the z-scores for the range limits and then use the normal table to find the area under the curve between those z-scores.

Question 27

What are the steps to find probabilities using the normal distribution?

Answer 27

1. Formulate the problem in terms of the random variable x.
2. Calculate relevant z-values and restate the problem in terms of z.
3. Find the area under the standard normal curve using the normal table.
4. Illustrate the area needed before using the normal table.

Question 28

How do you evaluate evidence against a claim using the normal distribution?

Answer 28

To evaluate evidence against a claim using the normal distribution, calculate the z-score for the observed value, then use the normal table to find the probability of obtaining that value or one more extreme if the claim is true.

Question 29

What does a z-score of -2.57 indicate about a car’s mileage in comparison to the claimed mean?

Answer 29

A z-score of -2.57 indicates that the car’s mileage is 2.57 standard deviations below the claimed mean mileage.

Question 30

How do you find the probability that a car’s mileage is less than or equal to 31.2 mpg?

Answer 30

To find the probability that a car’s mileage is less than or equal to 31.2 mpg, calculate the z-score and then use the normal table to find the cumulative probability to the left of the z-score.

Question 31

What does a probability of 0.0051 indicate in the context of car mileage?

Answer 31

A probability of 0.0051 indicates that if the competing automaker’s claim is valid, only about 51 in 10,000 cars would get a mileage of less than or equal to 31.2 mpg.

Question 32

How do you calculate the probability of a car’s mileage being within a certain range?

Answer 32

Calculate the z-scores corresponding to the range boundaries and then find the area under the standard normal curve between these z-scores using the normal table.

Question 33

How do you find the probability that a cup of coffee is either too hot or too cold?

Answer 33

Calculate the z-scores for temperatures too hot and too cold, then add the probabilities from the normal table for temperatures below and above the ideal range.

Question 34

What does z_(0.025) represent in a standard normal distribution?

Answer 34

z_(0.025) represents the z-value on the horizontal axis under the standard normal curve that gives a right-hand tail area equal to 0.025.

Question 35

How do you find the z-value that corresponds to a specified right-hand tail area?

Answer 35

To find a z-value for a specified right-hand tail area, subtract the tail area from 1 to find the cumulative area to the left, then look up this value in the body of the normal table.

Question 36

How can you determine the number of products to stock to ensure a specified probability of demand being met?

Answer 36

Determine the z-value that corresponds to the desired tail probability, then solve for the number of products using the z-score formula.

Question 37

How do you find a z-value for a left-hand tail area of 0.025?

Answer 37

Look for the area closest to 0.025 in the normal table and use the negative of the corresponding z-value from the right-hand tail, which is -1.96.

Question 38

How is the z-value for a specified left-hand tail area found?

Answer 38

The z-value is found by looking up the corresponding area in the normal table and identifying the z-value that gives the cumulative area up to that point.

Question 39

When is halfway interpolation used in finding z-values?

Answer 39

Halfway interpolation is used when the area we’re looking for is exactly halfway between two areas in the normal table.

Question 40

When can the normal distribution be used to approximate the binomial distribution?

Answer 40

When np ≥ 5 and n(1 - p) ≥ 5.

Question 41

What is the mean (μ) and standard deviation (σ) used in the normal approximation of the binomial distribution?

Answer 41

Mean μ = np and standard deviation σ = √(npq), where q = 1 - p.

Question 42

What is the continuity correction and why is it used?

Answer 42

The continuity correction is used because the normal distribution is continuous, and it adjusts for the approximation of a discrete binomial distribution by assigning a continuous area under the normal curve to discrete outcomes.

Question 43

How do you calculate the normal probability for a binomial random variable with 50 trials and a probability of success of 0.5 to find the probability of exactly 23 successes?

Answer 43

Use the continuity correction by assigning an area under the normal curve to the interval from 22.5 to 23.5, and calculate P(-.71 ≤ Z ≤ -.42), which is approximately 0.0983.

Question 44

How is the normal approximation applied when calculating the probability that the number of successes is within a certain range?

Answer 44

List the number of successes in the event, assign the areas under the normal curve corresponding to intervals adjusted by 0.5 for continuity correction, and sum the probabilities.

Question 45

How do you use the normal distribution to determine if a new process change is beneficial, given a sample result and the desired level of significance?

Answer 45

Calculate the z-value for the sample result and find the corresponding probability from the normal table. If the probability is less than the significance level, the new process is likely beneficial.