AP Statistics – Notes 7.1 Name: __________________ What is a Sampling Distribution? Today we will take a Sample from a Population, and we will use the sample mean to estimate the mean (or proportion) of the population. A Simple Example: My 4B Class is very small. There were only 4 people who took the Cumulative Quiz. Their scores were 60, 70, 80, and 90. 1. Make a dotplot of the population distribution. 2. Take a sample of any two of these scores. Find the mean of your sample and make a dotplot of your sample. 3. Figure out all possible samples of size 2. Calculate a statistic (sample mean) for each sample. 4. Make a dotplot using each of the statistics (sample means) you found in #3. 5. What is the mean of the population (Parameter)? Label this on your dotplot above. key Distribution of Population ooo J I l l l l l 50 Go 70 so go 100 Quiz scores Distribution of sample n 2 oooo oooo I 60 702 l l l l l l 2 50 Go 70 so go too 5 65 Quiz Scores n 60,20 05 65 no 70,80 0 4 75 na _60,80 0 Ia 70 n 70,90 0 5 80 n 60,90 05 75 n 80,90 05 85 MI 75 sampling Distribution of I A sample of 2 with a mean of65 J I l l l l l 50 Go 70 so go too 5 of all possible samples n 2 µ 60 20 80 90 4 U 75 Population mean
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AP Statistics – Notes 7.1 Name: __________________ What is a Sampling Distribution?
Today we will take a Sample from a Population, and we will use the sample mean to estimate the mean (or proportion) of the population. A Simple Example: My 4B Class is very small. There were only 4 people who took the Cumulative Quiz. Their scores were 60, 70, 80, and 90.
1. Make a dotplot of the population distribution.
2. Take a sample of any two of these scores. Find the mean of your sample and make a dotplot of your sample.
3. Figure out all possible samples of size 2. Calculate a statistic (sample mean) for each sample.
4. Make a dotplot using each of the statistics (sample means) you found in #3.
5. What is the mean of the population (Parameter)? Label this on your dotplot above.
key
Distribution of Population
ooo JI l l l l l50 Go 70 so go 100
Quizscores
Distribution of samplen 2 oooo oooo I60702 l l l l l l
2 50 Go 70 so go too5 65 QuizScores
n 60,20 05 65 no 70,80 0 4 75
na_60,80 0 Ia 70 n 70,90 0 5 80
n 60,90 05 75 n 80,90 05 85
MI 75 samplingDistribution of IAsampleof2withameanof65 JI l l l l l
50 Go 70 so go too5ofallpossiblesamples n 2
µ 60 20 80 904
U 75 Populationmean
AP Statistics – Notes 7.1 Name: __________________ What is a Sampling Distribution?
1. Distinguish between a parameter and a statistic.
Identify the population, the parameter, the sample, and the statistic in each of the following settings.
a) The Gallup Poll asked a random sample of 515 U.S. adults whether or not they believe in ghosts. Of the respondents, 160 said “yes”.
b) During the winter month, the temperatures outside the Starnes’s’ cabin in Colorado can stay well below freezing (32 Celsius or 0 Fahrenheit) for weeks at a time. To prevent the pipes from freezing, Mrs. Starnes sets the thermostat at 50 Fahrenheit. She wants to know how low the temperature actually gets in the cabin. A digital thermometer records the indoor temperature at 20 randomly chosen times during a given day. The minimum reading is 38 Fahrenheit.
A parameter is a number that describes some characteristic of the population.
A statistic is a number that describes some characteristic of a sample.
Key
statistics areused toestimateParametersstatistics Parameters
I o Mf o ps o 0
median 0Median
Parameter P ProportionofU.sAdultspopulation AllU.SAdults whobelieve in ghostssample 515us Adults stash D 5 0.3107
Population Temperatures at all times parameter minimumtemp induring agivenday cabin thatday
20 randomlyselected statistic minimum of 38 Ftemperaturereadings
AP Statistics – Notes 7.1 Name: __________________ What is a Sampling Distribution?
2. Distinguish among the distribution of a population, the distribution of a sample, and the sampling distribution of a statistic.
How many blue beads are in the jar?
**BEADS ACTIVITY DIRECTIONS**
1. Take a Random Sample of about 25 beads. 2. Record your statistics ( and n ) and graph your sampled data.
= ________ n = _________
3. Plot your on the board. 4. Copy the distribution from the board on to your notes.
Sampling Distribution of !"
Beads Activity Questions:
a) What does each dot on the board represent?
b) What do you think the true proportion of blue beads in the jar is?
c) A Sampling Distribution of a Statistics is the distribution of statistics in all possible samples of the same size from the same population. Is the dotplot above a sampling distribution? Explain
p̂
p̂
p̂
Key
26y 0.63 41
Fpf pp p pl i i e i l i y i r i e is iF i r j i is2
Proportion ofbluebeads in samples
Theproportionofbluebeadsin a sample
Mp 0.5
No because it does not include a statisticfromallpossible samples andall of thesesamples are notthe same size
AP Statistics – Notes 7.1 Name: __________________ What is a Sampling Distribution?
3. Use the sampling distribution of a statistic to evaluate a claim about a parameter.
d) Suppose we took a random sample of 25 beads, from a different jar (Call it Jar 2), and got a sample proportion of #̂ = 0.48. Is this convincing evidence that Jar 2 has a higher proportion of blue beads than Jar 1? Explain
4. Determine whether or not a statistic is an unbiased estimator of a population parameter. 5. Describe the relationship between sample size and the variability of a statistic.
Larger Samples = Less Variability
n = 100 n = 1000
A statistic used to estimate a parameter is an unbiased estimator if the mean of its sampling distribution is equal to the true value of the parameter being estimated.
AP Statistics – Notes 7.1 Name: __________________ What is a Sampling Distribution?
Example Problem:
The histogram above left shows the intervals (in minutes) between eruptions of Old Faithful geyser for all 222-recorded eruptions during a particular month. For this population, the median is 75 minutes. We used Fathom software to simulate taking 500 SRS’s of size 10 from the population. The 500 values of the sample median are displayed in the histogram above right. The mean of these 500 values is 73.5.
1. Is the sample an unbiased estimator of the population mean? Justify your answer.
2. Suppose we had taken samples of size 20 instead of size 10. Would the spread of the sampling distribution be larger, smaller, or about the same? Justify your answer.
Bias, Variability, and Shape We can think of the true value of the population parameter as the bull’s- eye on a target and of the sample statistic as an arrow fired at the target.
• Bias means that our aim is off and we consistently miss the bull’s-eye in the same direction. Our sample values do not center on the population value.
• High variability means that repeated shots are widely scattered on the target. Repeated samples do not give very similar results.
Nobecause themean of the approximatesamplingdistribution ofthesamplemedian 73.5 is notequal tothemedian of thepopulation 75
Increasing thesample size willdecreasethespread of thesamplingdistribution Largersamplesprovidemorepreciseestimatesbecausetheyincludemoreinformationaboutthepopulation
Largersamplesmeans moreprecisebut notnecessarilymoreaccurate
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