Biostatistics course Part 8 Inferences of a mean

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Biostatistics course Part 8 Inferences of a mean. Dr. Sc Nicolas Padilla Raygoza Department of Nursing and Obstetrics Division of Health Sciences and Engineering Campus Celaya Salvatierra University of Guanajuato Mexico. Biosketch. Medical Doctor by University Autonomous of Guadalajara.

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### Biostatistics coursePart 8Inferences of a mean

Department of Nursing and Obstetrics

Division of Health Sciences and Engineering

Campus Celaya Salvatierra

University of Guanajuato Mexico

Biosketch

Medical Doctor by University Autonomous of Guadalajara.

Pediatrician by the Mexican Council of Certification on Pediatrics.

Postgraduate Diploma on Epidemiology, London School of Hygine and Tropical Medicine, University of London.

Master Sciences with aim in Epidemiology, Atlantic International University.

Doctorate Sciences with aim in Epidemiology, Atlantic International University.

Professor Titular A, Full Time, University of Guanajuato.

Level 1 National Researcher System

Competencies
• The reader will apply a Z test to obtain inferences of a mean.
• The reader will obtain a confidence interval for a mean.
• He (she) will apply a t test for a mean in a short sample.
• He (she) will obtain a confidence interval for a mean in a short sample.
Introduction
• If we measure the stature of students of FEOC, we can obtain its mean and standard deviation:
• Number of students: 269
• Mean of stature: 161.6 cm
• Standard deviation: 6.3 cm
• Median: 159 cm
• Range: 149 a 185 cm.
Notation
• For parameters of population, we use Greek letters; to parameters in sample, we use Roman letters.
Sampling distribution
• If we take many samples of the same size of the same population, each sample can have different mean and standard deviation.
• If we plot these sample means we can obtain a sampling distribution.
• If the sample size is big, the mean distribution is almost Normal, although data distribution in the population is not Normal.
Sampling distribution (contd…)

Data of students from FEOC. If we take other 999 samples of students, we can graphic the distribution of their means.

95% Confidence Interval
• They use the probability theory to obtain conclusions on a population, from data obtained of a sample.
• It is difficult study all population, because of this, we study samples.
• Methods for obtain estimates and hypothesis test are important to obtain inferences.
95% confidence intervals (contd…)
• Then, the confidence intervals for a mean, are calculated:

_

X ± 1.96 (ES)

_

X is the estimate obtained of the sample,

1.96 is the multiply of standard errors for 95%,

SE is the standard error

• We should wait that the 95% confidence interval around of the mean of sample include the mean of the population in the 95% of times, if we obtain thousands of samples.
95% confidence intervals (contd…)
• We calculate 95% confidence interval for the first sample of 269 students from FEOC:

_

X = 161.6

SE= 6.3/√269= 0.38

95%CI= 161.6 ± 1.96 (0.38) = 161.6 ± 0.74 =

160.86 a 162.34

95% confidence intervals (cont…)
• We can use confidence intervals in another percentage of confidence, only we need change the multiply of standard error:
• For example, for 90% change to 1.69.
• For 95.4% change to 2.
• For 99% change to 3.
Hypothesis test for a mean
• Hypothesis test is to probe if our estimate is similar with a specific value.
• Our sample of 269 students had a mean of 161.6 with standard deviation of 6.3 and standard error of 0.38.
• In a similar study in students from School of Accounting and Administration, obtained a mean of stature of 167 cm.
• How we can demonstrate if the stature of students from FEOC is equal or different that stature of students from FCA?
• Mean of FEOC 161.6
• Mean of FCA 167
• We can see that obviously, they are different.
• But, we do not know if the observe difference is true or it is by error sampling, because 161.6 is an estimate of many that we can have obtaining
Hypothesis test for a mean (contd…)
• To evaluate if the observe difference is real, we can do:
• Null hypothesis say that the means of both populations are the same (the first population is students from FEOC and reference population is students from FCA).
• Null hypothesis is writing as Ho.
• If the mean of hypothesis is μo and the mean in study is μ, then, null hypothesis is writing as HO : μ = μo
• Alternative hypothesis
• It is that the means of two populations are not equal.
• Usually, it is writing as H1: μ≠μ0
Hypothesis test for a mean (contd…)
• When we are pointing the null hypothesis, calculate the probability of obtain the observe data if the null hypothesis is true.
• To obtain this probability, we calculate a statistic test and it is compare with the distribution implicated for the null hypothesis.
• In many cases it will be Normal distribution.
Hypothesis test for a mean (contd…)
• The general form of statistic test compare the estimate of observed values of the sample and the expected value if the null hypothesis is true.
• Also, it take into account the variability in the population using standard error.
• This statistic test is called Z and it is equal to:

_

X – μo

Z= ------------

ES

_

Then, the test is a standardized difference between X and μo.

Example
• The students sample from FEOC

Mean = 161.6

S = 6.3

95%CI = 160.6 a 162.60

• Null hypothesis; there is not difference between the means of students from FEOC and FCA

Ho: μ = 167cm

We need use Z test:

_

X – μo 161.6 -167

z = ----------- = ---------------- = - 14.21

ES(X) 0.38

Small samples
• If the sample size is small, we use t distribution.
• Its form depend of freedom degrees, that it is a measure that is so small is sample size.
• The degree freedom of a t distribution is equal to sample size minus 1.
Small samples
• Less freedom degrees, less probability of stay around of mean of sample and high probability to stay in the tails.
• The t distributions with a few freedom degrees have more smaller probabilities to sides of the mean and higher probabilities in the tails.
• However, if the samples size is bigger and more freedom degrees, more similar is t distribution to Normal distribution.
• There are published tables of selected values of the area under t distribution that we shall use when calculate confidence intervals and hypothesis test.
Small samples
• When the sample size is small, less than 100, the formulas for confidence interval and hypothesis test, are:

95%CI Hypothesis test

Estimate ± multiplier (standard error) To test Ho: μ=μo

Estimate is the estimate mean To test H1: μ≠μo

Multiplier is the value of t _

Correspond to p=0.05 with degree X – μ0

Freedom equal to sample size minus 1 t = ---------

SE

P values
• One or two tails?
• Now, we know that the p value is the probability to obtain a result at least extreme as the found with our sample, if the null hypothesis are true.
• But, what is the meaning of extreme?
• When the alternative hypothesis is H1: µ ≠ µo
• Then, the extreme results can occur for chance at each side of the mean of the hypothesis, µo.
• Due of this, we used tables for two tails of Normal and t distributions.
P values
• There are occasions less common where the alternative hypothesis is H1: µ < µo or H1: µ > µo
• Then, extreme values can occur only to the left or only to the right, of the mean of hypothesis.
• How little is little the p value?
• Many people are using the p value of 0.05 as cut point. This is a arbitrary value, but it is sensitive. The meaning is that we are prepare to reject the null hypothesis at least one time of 20 when is true.
• Note that when the value of a test has a p value less than 0.05, the confidence interval does not include the hypothesis value.
P values
• If we obtained a p value of 0.048, can we reject the null hypothesis?
• If we obtained a p value of 0.052 , do we cannot reject the null hypothesis?
• When the p values are between 0.07 and 0.03 they should be joint the real p value, because they are in the border of significance statistic.
Showing the results
• We should show the results with their confidence intervals.
• Clear what is the null and alternative hypothesis.
• Show the p value of each test; it is sufficient with say p< 0.001 when apply.
• Not misunderstood the p values
• A small p value reject the null hypothesis,
• A high p value only does not reject the null hypothesis
Bibliografía
• 1.- Last JM. A dictionary of epidemiology. New York, 4ª ed. Oxford University Press, 2001:173.
• 2.- Kirkwood BR. Essentials of medical statistics. Oxford, Blackwell Science, 1988: 1-4.
• 3.- Altman DG. Practical statistics for medical research. Boca Ratón, Chapman & Hall/ CRC; 1991: 1-9.