Central Limit Theorem

Question 1

Central Limit Theorem

Answer 1

• As you take more samples, especially large ones(more than 30), your graph of the same means (averages) will look more like a normal distribution.
• The sample mean distribution of a random variable will assume a near-normal or normal distribution if the same size is large enough.
  
  • The theorem states that the sampling distribution of the mean
• The distribution of the same means approaches normal regardless of the shape of the parent population
• Same means (s) will be normally more distributed around (µ) than the individual readings (Xs).
• As n - the sample size - increases, then the sample averages (Xs means) will approach a normal distribution with mean(µ).

Question 2

Central Limit Theorem Formula

Answer 2

z= x̅- µ/ σ/√n

Use Z-Score

µ = population mean

σ = population standard deviation

n = sample size

x̅ = sample mean

Question 3

Significance of Central Limit Theorem

Answer 3

• The central limit theorem is one of the most profound and useful results in all statistics and probability.
  
  • The large samples (more than 30) from any sort of distribution the same means will follow a normal distribution.
• The spread of the sample means is less (narrower) than the spread of the population you’re sampling from. So, it does not matter how the original population is skewed
  
  • The means of the sampling distribution of the mean is equal to the population mean:
    
    • µ_x̅ =µ_X
  • The standard deviation of the sample means equals the standard deviation of the population divided by the square root of the sample size:
    
    • σ(_x̅) = σ(x) / √(n)

Question 4

Central Limit Theorem Assumptions

Answer 4

• Samples must be independent of each other
• Samples follow random sampling
• If the population is skewed or asymmetric, the sample should be sufficiently large (for example minimum of 30 samples)

Question 5

Why Central Limit Theorem is Important

Answer 5

• The Central Limit Theorem allows the use of confidence intervals, hypothesis testing, DOG, regression analysis, and other analytical techniques.
• Many statistics have approximately normal distributions for large sample sizes, even when we are sampling from a distribution that is non-normal
  
  • This means that we can often use well developed statistical inference procedures and probability calculations that are based on a normal distribution, even if we are sampling from a population that is not normal, provided we have a large sample size.

Question 6

Unimodal Distribution

Answer 6

• Unimodal Distribution will have only one peak or only one frequent value in the data set.
• Unimodal will have only one mode, the values are increases first and reaches to peak (e.i. is the model or the local maximum) and the decreases.
  
  • Mode is one of the measures of central tendency. Mode is the value that appears most often in a set of data values or a frequent number
• Normal Distribution is the best example of Unimodal.
  
  • Bimodal distribution means there are two different modes, and multimodal means more than two different modes.

Question 7

Central Limit Theorem Example

Less Than

Answer 7

• A population of 65 years male patient blood sugar was 100 mg/dL with a standard deviation of 15 mg/Dl. If a sample of 4 patients’ data were drawn, what is the probability of their mean blood sugar is less than 120 mg/dL?
  
  • µ = 100
  • x̅ = 120
  • n=4
  • σ =15
    
    • z= x̅- µ / σ/√n
    • =120-100/15/√4
    • =20/7.75
    • =2.66
      
      • Look up on the Z table z=2.6 on left and z=0.06 on top
        
        • =0.9961
        • =99.61%

Question 8

Central Limit Theorem Example

Between

Answer 8

• A population of 65 year old male patients blood sugar was 100 mg/dL with a standard deviation of 20 mg/dL. If a sample of 9 patients’ data were drawn, what is the probability of their mean blood sugar is between 85 and 105 mg/dL?
• First, compute P(x<105)
  
  • µ = 100
  • x̅ = 105
  • n=9
  • σ =20
    
    • Compute the Z score z= x̅- µ/ σ/√n
    • = 105-100/20/√9
    • =5/6.67
    • =0.75
      
      • On Z-Table find z=0.7 on left and z=0.05 on top
        
        • =0.7734
        • =77.3%
  • Second, compute P(x<85)
    
    • µ = 100
    • x̅ = 85
    • n=9
    • σ =20
      
      • Compute the Z score z= x̅- µ/ σ/√n
        
        • = 85-100/20/√9
        • = -15/6.67
        • = -2.24
          
          • On negative Z-Table find z= - 2.2 on left and z=0.04 on top
          • = 0.0125
          • = 1.25%
  • Third, since we are looking for blood sugar between 85 and 105 mg/dL P(85
  • = 77.3-1.25
  • = 76.05%
• Answer: The probability of mean blood sugar is between 85 and 105 mg/dL is 76.05%

Question 9

Central Limit Theorem Example

Greater Than

Answer 9

• A population of 65 year old made patients blood sugar was 100 mg/dL with a standard deviation of 20 mg/dL. If a sample of 16 patients’ data were drawn, what is the probability of their mean blood sugar is more than 90 mg/dL?
  
  • µ = 100
  • x̅ = 90
  • n=16
  • σ =20
    
    • Compute the Z score z= x̅- µ/ σ/√n
      
      • = 90-100/20/√16
      • = -10/5
      • = -2
        
        • In Z table locate z -2.0 on left side and z 0.00 on top
          
          • = 0.0228
          • = 2.27%
    • Since we are looking for blood sugar more than 90 mg/dL
      
      • = 100% - 2.28%
      • = 97.72%
      • Answer: The probability of mean blood sugar is greater than 90 mg/dL is 97.72%
        *

Question 10

Confidence Intervals

Answer 10

• Confidence intervals are used to calculate a degree of certainty that the sample group accurately represents the entire population from which they are drawn.
  
  • Another way of thinking of it is that if you drew the same sized sample group hundreds of times and performed the same measurements, a certain percentage of confidence internals in those sample groups will contain the population mean
• A confident interval is a range of values.
  
  • You have a percentage of certainty that the mean of the population lies within that range in any given sample from that population.

Question 11

The x-bar control charts makes use of what statistical concept?

Answer 11

• The Central Limit Theorem
  
  • The Central Limit Theorem is the concept that states that the average values of samples drawn for any statistical universe, regardless of the shape of the distribution of the population of that universe, will tend toward a normal distribution as the sample size grows

Question 12

Which distribution describes the probability of r occurrences in n trails of an event?

Answer 12

• Binomial

Question 13

Answer 13