okay so I'm taking a college level basic introduction to statistics course to fulfill a math requirment before moving on with my usual array of literature and theatre courses. And I've been doing okay the last couple weeks using the formulas they give me and what have you, but now I'm getting questions like this and I literally have no idea what to do with them to even attempt to solve them.
The question:
"Suppose the wrapper of a candy bar lists its weight as 8 ounces. The actual weights of individual candy bars naturally vary to some extent, however. Suppose these actual weights vary according to a normal distribution with mean 8.5 ounces and standard deviation .325 ounces.
a)what proportion of the candy bars weigh less than the advertised 8 ounces?
b) If the manufacturer wants to decrease this proportion from a) by changing the mean of its candy bar weights, should it increase or decrease that mean? Explain briefly without using calculations.
c) If the manufacturer wants to decrease this proportion from a) by changing the standard deviation of its candy bar weights, should it increase or decrease that standard deviation? Explain briefly, without performing any calculations."
So I look at this and just with part a) I have no idea what to do or where to even begin to look for information on what to do (hence a google search leading me here). I look at three major distributions we worked with in class (Binomial, Geometric and Poisson) and as I understand them none of them seem to apply to this. I see the word Normal Distribution in the question and go to the chapter in the textbook with that name but I don't see any kind of formula that helps me (in fact all I see is stuff about drawing some curve line on a chart and how to turn z into x--whatever that means). I look at the Central Limit Theorem because that's the most recent thing we discussed in class and the only formula I see in that chapter is the Confidence Interval for Population Proportion thing: but that involves solving E with a square root that involves p, q and n and I don't see to have enough numbers in the word problem to plug into that many variables (specifically I can't find any value that would represent n). Which leaves me at a complete loss, so any help on what I need to do would be peachy keen.
If it helps so we have a common frame of reference these are my notes from the last three chapters that we talked about in class (the formulas themselves are on a reference sheet so I rarely put them into my notes). Mind you just cause something is in my notes does not necessarily mean I understand it, it just means it seemed important so I wrote it down. Unfortunately posting it here takes out all the bolding and italicizing so it probably reads like massive run-on sentence without that.
chapter 4
In a random variable x represents a numerical value associated with each outcome of a probability experiment. A random variable is discrete if it has a finite or countable number of possible outcomes that can be listed. A random variable is continuous if it has an uncountable number of possible outcomes, represented by an interval on the number line. In most practical applications discrete random variables represent count data, while continuous random variables represent measured data. Values of variables such as age, height and weight are usually rounded to the nearest year, inch or pound. However, these values represent measured data, so they are continuous random variables.
A discrete probability distribution lists each possible value the random variable can assume, together with its probability. A probability distribution must satisfy the following conditions: 1-The probability of the value of the discrete random variable is between 0 and 1, inclusive; 2- The sum of all probabilities is 1. To construct a discrete probability distribution: 1 - Make a frequency distribution chart for the possible outcomes; 2- Find the sum of the frequencies; 3- Find the probability of each possible outcome by dividing its frequency by the sum of the frequencies, 4- Check that each probability is between 0 and 1 and thus the sum is 1.
To find the mean of a discrete random variable take each value of x and multiply it by its corresponding probability and then add all the products (results of multiplication). Note there is no division here ?
The expected value of a discrete random variable is equal to the mean of the random variable (formula sheet).
Binomial Distributions
1. Fixed number of trials, independent of each other.
2. Only two possible outcomes: success(S) or failure(F)
3. Probability of Success P(S) is the same for each trial.
4. x counts the number of successful trials
Notations for this stuff
• N = the # of trials reported
• P(S) = probability of success
• Q = probability of failure / also P(F)
• x = number of successful trials
• a big capital C = combination – the order of variables does not matter.
see chart
Recall that if a probability is 0.05or less it is considered unusual
Geometric Distribution
1. A trial is repeated until success occurs (thus the last entry is always a success and all prior entries are failures)
2. The repeated trials are independent of each other.
3. When calculating the probability of success the P is constant. (see chart)
Poisson Distribution
1. The experiment consists of counting the number of times x (an event) occurs over a specified interval of time, area or volume.
2. The probability of an event occurring is the same for each interval.
3. The number of outcomes is independent see chart
Note the formula contains a negative exponent. To calculate this do the normal exponent multiplication and divide the result into 1. Thus 2 to -3 would be2*2*2 (so 8) and then 1 / 8.
Teacher tips:
• To find the mean of a distribution multiply each example of x (what is being measured such as # of computers) by its probability of happening and then add all the products. Then use that mean in standard deviation formula.
• As a sample size gets bigger the more symmetric it becomes when graphed (think rolling 2 dice and prob of 7 vs 2 or 12)
• Any exponent to 0 makes the # it is applied to become 1
• In the above distributions to find probability multiply the prob of each column until the # going to. So if prob success is 19% and we want to know prob on 3rd try it would be: .81 * .81 * .19. If it is multiple numbers using or repeat the process for each number and add so chance of 1, 2 or 3 = .19 + (.81 * .19) + (.81 * .81 * .19).
• If using the word at least remember the sum of all probs = 1 so thus at least 5 means P(0,1,2,3,4) + P(At least 5) = 1. Thus solve first half and subtract from 1 to find At Least.
chapter 5
Properties of a Normal Distribution
A normal distribution is a continuous probability distribution for a random variable x. The graph of a normal distribution is called the normal curve. A normal distribution has the following properties:
1. The mean, median and mode are equal
2. The normal curve is bell-shaped and symmetric about the mean
3. The total area under the normal curve is equal to one
4. The normal curve approaches, but never touches, the x-axis as it extends farther and father away from the mean.
5. Between the mean – standard deviation and mean + standard deviation (in the center of the curve), the graph curves downward. The graph curves upward to the left of m-o and the right of m+o. The points at which the curve changes from curving upward to curving downward are called inflection points.
To learn how to determine if a random sample is taken from a normal distribution see Appendix C
The normal distribution with a mean of 0 and a standard deviation of 1 is called the standard normal distribution. Because every normal distribution can be transformed to the standard normal distribution, you can use z-scores and the standard normal curve to find areas (and therefore probability) under any normal curve. The random variable x is sometimes called a raw score and represents values in a nonstandard normal distribution, whereas z represents values in the standard normal distribution. To transform an x-score into a z-score use the formula z = x – mean / standard deviation.
Properties of the Standard Normal Distribution
1. The cumulative area is close to 0 for z-scores close to z = (-3.49)
2. The cumulative area increases as the z-score increases.
3. The cumulative area for z = 0 to 0.5000
4. The cumulative area is close to 1 for z-scores close to z = 3.49
Finding Areas Under the Standard Normal Curve
• To find the area left of z, find the area that corresponds to z in the standard normal table.
• To find the area to the right of z, use the standard normal table to find the area that corresponds to z then subtract the area from 1.
• To find the area between two z-scores, find the area corresponding to each z-score in the standard normal table. Then subtract the smaller area from the larger area.
percents on 262 ???
To transform a standard z-score to a data value x use the formula x = mean + z *standard deviation.
A sampling distribution is the probability distribution of a sample statistic that is formed when samples of size n are repeatedly taken from a population. If the sample statistic is the sample mean, then the distribution is the sampling distribution of a sample means. Every sample statistic has a sampling distribution.
Properties of sampling distribution of a sample means
1. The means of the sample is equal to the population mean
2. The standard deviation of the sample is equal to the population standard deviation / square root of n
The standard deviation of the sampling means is called the standard error of the mean.
The Central Limit Theorem
1. If samples of size n, where n > 30, are drawn from any population with a mean and standard deviation, then the sampling distribution of sample means approximates a normal distribution. The greater the sample size, the better the approximation.
2. If population itself is normally distributed, the sampling distribution of sample means is normally distributed for any sample size n.
Normal Approximation to a Binomial Distribution
If np > 5 and nq > 5, then the binomial random variable x is approximately normally distributed with a mean = np and a standard deviation equal to the square root of npq.
1. Verify the binomial distribution (mp, q)
2. can you approximate normally (np>5, nq>5
3. Find mean, standard deviation (m=np, o- = v/npq)
4 Apply continuity correction (+/- .05)
5 Draw curve and shade
6 Find Z (Z = x-overline x / o-)
7 Find probability
chapter 6
confidence intervals for the mean (large samples): Point Estimate – single value estimate for a parameter in m and x- are the same (the mean) and o- and s are the same (standard dev). Interval estimate – range of values to estimate a parameter. Level of Confidence (c) – probability that the interval estimate contains the parameter (these are fixed #s for Zc); 90%=1.645, 95%=1.96, 99%=2.575. (see chart for margin of error and for confidence interval for population mean, the latter tells us if a large # of sample is collected and a confidence interval is created for each sample, approximately C of these intervals will contain m. confidence intervals for the mean (small samples): relates to T-thing on table 5. Confidence intervals for population proportions are on chart.
leftovers
How to find Standard Deviation
1. Find the sample mean (the average of the table, aka the sum of all entries divided by the # of entries).
2. Subtract the sample mean from every entry in the set.
3. Square each difference (the result of the subtraction)
4. Add the Squares
5. Divide by the sample size – 1 (so one less than the number you divided by in step 1).
6. Take the square root of the result of Step 5 and that is the Standard Deviation
The Empirical Rule then tells us that in spherical tables 68% of the data will be within one Standard Deviation, 95 % within two Standard Deviation and 99.7 % within three Standard Deviations.
Z-score – is the number of Standard Deviations above or below the mean of a single data entry in a table. To find it choose the entry you want to measure and subtract the mean from it then divide the result by the Standard Deviation.
The question:
"Suppose the wrapper of a candy bar lists its weight as 8 ounces. The actual weights of individual candy bars naturally vary to some extent, however. Suppose these actual weights vary according to a normal distribution with mean 8.5 ounces and standard deviation .325 ounces.
a)what proportion of the candy bars weigh less than the advertised 8 ounces?
b) If the manufacturer wants to decrease this proportion from a) by changing the mean of its candy bar weights, should it increase or decrease that mean? Explain briefly without using calculations.
c) If the manufacturer wants to decrease this proportion from a) by changing the standard deviation of its candy bar weights, should it increase or decrease that standard deviation? Explain briefly, without performing any calculations."
So I look at this and just with part a) I have no idea what to do or where to even begin to look for information on what to do (hence a google search leading me here). I look at three major distributions we worked with in class (Binomial, Geometric and Poisson) and as I understand them none of them seem to apply to this. I see the word Normal Distribution in the question and go to the chapter in the textbook with that name but I don't see any kind of formula that helps me (in fact all I see is stuff about drawing some curve line on a chart and how to turn z into x--whatever that means). I look at the Central Limit Theorem because that's the most recent thing we discussed in class and the only formula I see in that chapter is the Confidence Interval for Population Proportion thing: but that involves solving E with a square root that involves p, q and n and I don't see to have enough numbers in the word problem to plug into that many variables (specifically I can't find any value that would represent n). Which leaves me at a complete loss, so any help on what I need to do would be peachy keen.
If it helps so we have a common frame of reference these are my notes from the last three chapters that we talked about in class (the formulas themselves are on a reference sheet so I rarely put them into my notes). Mind you just cause something is in my notes does not necessarily mean I understand it, it just means it seemed important so I wrote it down. Unfortunately posting it here takes out all the bolding and italicizing so it probably reads like massive run-on sentence without that.
chapter 4
In a random variable x represents a numerical value associated with each outcome of a probability experiment. A random variable is discrete if it has a finite or countable number of possible outcomes that can be listed. A random variable is continuous if it has an uncountable number of possible outcomes, represented by an interval on the number line. In most practical applications discrete random variables represent count data, while continuous random variables represent measured data. Values of variables such as age, height and weight are usually rounded to the nearest year, inch or pound. However, these values represent measured data, so they are continuous random variables.
A discrete probability distribution lists each possible value the random variable can assume, together with its probability. A probability distribution must satisfy the following conditions: 1-The probability of the value of the discrete random variable is between 0 and 1, inclusive; 2- The sum of all probabilities is 1. To construct a discrete probability distribution: 1 - Make a frequency distribution chart for the possible outcomes; 2- Find the sum of the frequencies; 3- Find the probability of each possible outcome by dividing its frequency by the sum of the frequencies, 4- Check that each probability is between 0 and 1 and thus the sum is 1.
To find the mean of a discrete random variable take each value of x and multiply it by its corresponding probability and then add all the products (results of multiplication). Note there is no division here ?
The expected value of a discrete random variable is equal to the mean of the random variable (formula sheet).
Binomial Distributions
1. Fixed number of trials, independent of each other.
2. Only two possible outcomes: success(S) or failure(F)
3. Probability of Success P(S) is the same for each trial.
4. x counts the number of successful trials
Notations for this stuff
• N = the # of trials reported
• P(S) = probability of success
• Q = probability of failure / also P(F)
• x = number of successful trials
• a big capital C = combination – the order of variables does not matter.
see chart
Recall that if a probability is 0.05or less it is considered unusual
Geometric Distribution
1. A trial is repeated until success occurs (thus the last entry is always a success and all prior entries are failures)
2. The repeated trials are independent of each other.
3. When calculating the probability of success the P is constant. (see chart)
Poisson Distribution
1. The experiment consists of counting the number of times x (an event) occurs over a specified interval of time, area or volume.
2. The probability of an event occurring is the same for each interval.
3. The number of outcomes is independent see chart
Note the formula contains a negative exponent. To calculate this do the normal exponent multiplication and divide the result into 1. Thus 2 to -3 would be2*2*2 (so 8) and then 1 / 8.
Teacher tips:
• To find the mean of a distribution multiply each example of x (what is being measured such as # of computers) by its probability of happening and then add all the products. Then use that mean in standard deviation formula.
• As a sample size gets bigger the more symmetric it becomes when graphed (think rolling 2 dice and prob of 7 vs 2 or 12)
• Any exponent to 0 makes the # it is applied to become 1
• In the above distributions to find probability multiply the prob of each column until the # going to. So if prob success is 19% and we want to know prob on 3rd try it would be: .81 * .81 * .19. If it is multiple numbers using or repeat the process for each number and add so chance of 1, 2 or 3 = .19 + (.81 * .19) + (.81 * .81 * .19).
• If using the word at least remember the sum of all probs = 1 so thus at least 5 means P(0,1,2,3,4) + P(At least 5) = 1. Thus solve first half and subtract from 1 to find At Least.
chapter 5
Properties of a Normal Distribution
A normal distribution is a continuous probability distribution for a random variable x. The graph of a normal distribution is called the normal curve. A normal distribution has the following properties:
1. The mean, median and mode are equal
2. The normal curve is bell-shaped and symmetric about the mean
3. The total area under the normal curve is equal to one
4. The normal curve approaches, but never touches, the x-axis as it extends farther and father away from the mean.
5. Between the mean – standard deviation and mean + standard deviation (in the center of the curve), the graph curves downward. The graph curves upward to the left of m-o and the right of m+o. The points at which the curve changes from curving upward to curving downward are called inflection points.
To learn how to determine if a random sample is taken from a normal distribution see Appendix C
The normal distribution with a mean of 0 and a standard deviation of 1 is called the standard normal distribution. Because every normal distribution can be transformed to the standard normal distribution, you can use z-scores and the standard normal curve to find areas (and therefore probability) under any normal curve. The random variable x is sometimes called a raw score and represents values in a nonstandard normal distribution, whereas z represents values in the standard normal distribution. To transform an x-score into a z-score use the formula z = x – mean / standard deviation.
Properties of the Standard Normal Distribution
1. The cumulative area is close to 0 for z-scores close to z = (-3.49)
2. The cumulative area increases as the z-score increases.
3. The cumulative area for z = 0 to 0.5000
4. The cumulative area is close to 1 for z-scores close to z = 3.49
Finding Areas Under the Standard Normal Curve
• To find the area left of z, find the area that corresponds to z in the standard normal table.
• To find the area to the right of z, use the standard normal table to find the area that corresponds to z then subtract the area from 1.
• To find the area between two z-scores, find the area corresponding to each z-score in the standard normal table. Then subtract the smaller area from the larger area.
percents on 262 ???
To transform a standard z-score to a data value x use the formula x = mean + z *standard deviation.
A sampling distribution is the probability distribution of a sample statistic that is formed when samples of size n are repeatedly taken from a population. If the sample statistic is the sample mean, then the distribution is the sampling distribution of a sample means. Every sample statistic has a sampling distribution.
Properties of sampling distribution of a sample means
1. The means of the sample is equal to the population mean
2. The standard deviation of the sample is equal to the population standard deviation / square root of n
The standard deviation of the sampling means is called the standard error of the mean.
The Central Limit Theorem
1. If samples of size n, where n > 30, are drawn from any population with a mean and standard deviation, then the sampling distribution of sample means approximates a normal distribution. The greater the sample size, the better the approximation.
2. If population itself is normally distributed, the sampling distribution of sample means is normally distributed for any sample size n.
Normal Approximation to a Binomial Distribution
If np > 5 and nq > 5, then the binomial random variable x is approximately normally distributed with a mean = np and a standard deviation equal to the square root of npq.
1. Verify the binomial distribution (mp, q)
2. can you approximate normally (np>5, nq>5
3. Find mean, standard deviation (m=np, o- = v/npq)
4 Apply continuity correction (+/- .05)
5 Draw curve and shade
6 Find Z (Z = x-overline x / o-)
7 Find probability
chapter 6
confidence intervals for the mean (large samples): Point Estimate – single value estimate for a parameter in m and x- are the same (the mean) and o- and s are the same (standard dev). Interval estimate – range of values to estimate a parameter. Level of Confidence (c) – probability that the interval estimate contains the parameter (these are fixed #s for Zc); 90%=1.645, 95%=1.96, 99%=2.575. (see chart for margin of error and for confidence interval for population mean, the latter tells us if a large # of sample is collected and a confidence interval is created for each sample, approximately C of these intervals will contain m. confidence intervals for the mean (small samples): relates to T-thing on table 5. Confidence intervals for population proportions are on chart.
leftovers
How to find Standard Deviation
1. Find the sample mean (the average of the table, aka the sum of all entries divided by the # of entries).
2. Subtract the sample mean from every entry in the set.
3. Square each difference (the result of the subtraction)
4. Add the Squares
5. Divide by the sample size – 1 (so one less than the number you divided by in step 1).
6. Take the square root of the result of Step 5 and that is the Standard Deviation
The Empirical Rule then tells us that in spherical tables 68% of the data will be within one Standard Deviation, 95 % within two Standard Deviation and 99.7 % within three Standard Deviations.
Z-score – is the number of Standard Deviations above or below the mean of a single data entry in a table. To find it choose the entry you want to measure and subtract the mean from it then divide the result by the Standard Deviation.