The weights of a large number of miniature poodles are...

Name: Problem 20, Chapter 6: Probability and Statistics for Engineers and Scientists
Uploaded: 2021-07-05T23:48:23-08:00
Duration: 4 min 59 s
Channel: Adam Harper
Description: The weights of a large number of miniature poodles are approximately normally distributed with a mean of 8 kilograms and a standard deviation of 0.9 kilogram. If measurements are recorded to the nearest tenth of a kilogram, find the fraction of these poodles with weights (a) over 9.5 kilograms; (b) of at most 8.6 kilograms; (c) between 7.3 and 9.1 kilograms inclusive.

The weights of a large number of miniature poodles are approximately normally distributed with a mean of 8 kilograms and a standard deviation of 0.9 kilogram. If measurements are recorded to the nearest tenth of a kilogram, find the fraction of these poodles with weights (a) over 9.5 kilograms; (b) of at most 8.6 kilograms; (c) between 7.3 and 9.1 kilograms inclusive.

The weights of a large number of miniature poodles are approximately normally distributed with a mean of 8 kilograms and a standard deviation of 0.9 kilogram. If measurements are recorded to the nearest tenth of a kilogram, find the fraction of these poodles with weights
(a) over 9.5 kilograms;
(b) of at most 8.6 kilograms;
(c) between 7.3 and 9.1 kilograms inclusive.

Key Concepts

Measurement Rounding and Continuity Correction

When measurements are recorded to a specific precision, such as the nearest tenth, there is an element of rounding involved. In statistical calculations with continuous distributions, a continuity correction may be applied to adjust the discrete measurement to the continuous scale. This correction accounts for the fact that the underlying data, although measured discretely, are assumed to be continuously distributed.

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) of the normal distribution gives the probability that a random variable is less than or equal to a specified value. It is used to compute probabilities for various ranges by finding the area under the normal curve up to a given point. For probabilities over or between values, properties of the CDF, including symmetry of the normal curve, are applied.

Standardization (Z-scores)

Standardization involves converting a normally distributed random variable to a standard normal variable (Z-score) by subtracting the mean and dividing by the standard deviation. This transformation allows the use of standard normal distribution tables or computational tools to calculate probabilities regardless of the specific mean and standard deviation of the original variable.

Normal Distribution

The normal distribution is a continuous probability distribution that is symmetric about its mean. It is characterized by its bell-shaped curve, where most of the data tend to cluster around the mean, and the spread of the data is determined by the standard deviation. This distribution is widely used in statistics to represent real-valued random variables whose distributions are not known but are assumed to be influenced by many small, independent effects.

Transcript

00:01 All right, so we have some miniature poodles that have a mean of 8 kilograms and a standard deviation of 0 .9 kilograms.

00:16 Since this is a normal distribution, we're going to be using the normal distribution formula, where a is the number of poodles minus the mean divide by the standard deviation.

00:28 Okay, and this formula is always assuming we are less than or equal to the number of poodles.

00:36 It gives us the percentage that you get a number of poodles less than or equal to a.

00:46 So for part a, we want to know about greater than 9 .5 kilograms.

00:53 The probability of getting over 9 .5 kilograms in a poodle.

01:01 All right, so this just gets plugged into our formula.

01:04 9 .5 minus 8 over .9 .9 .9.

01:13 That's 1 .5 divide by .9, which is a z value.

01:19 This inside part is the z, is a z value.

01:24 It gives a z value of 1 .67.

01:29 Now you need to take that and look up on a z table what the probability should be.

01:36 So 1 .67, that gives a percentage of 95 .25%.

01:51 But we want greater than.