Ttest Example

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T-test Example

• With a sample of 100 accounts, Emefa finds that
  the mean age of outstanding receivables is 52.5
  days, with a standard deviation of 14. Do these
  results indicate the population mean might be 50
  days?

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• In this problem, we have only the sample standard
  deviation (s). This must be used in place of the
  population standard deviation ( ).
• When we substitute s for ,
• we use the t distribution, especially if the
  sample size is less than 30.

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• This significance test is conducted by following
  the six-step procedure
• 1. Null hypothesis
• 2. Statistical test
• 3. Significance level
• 4. Calculated value
• 5. Critical test value
• 6. Interpret

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• 1. Null hypothesis.
• Ho 50 days
• HA gt 50 days (?-tailed test)
• 2. Statistical test.
• Choose the t-test because the data are ratio
  measurements
• Assume the underlying population is normal and we
  have randomly selected the sample from the
  population of customer accounts.

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• 3. Significance level.
• Let alpha .05, with n 100
• 4. Calculated value

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• 5. Critical test value.
• This is obtainable by entering the table of
  critical values of t
• With 99 degrees of freedom (d.f.) and a level of
  significance value of .05.
• The critical value of about 1.66 is obtained May
  be interpolated, if not exact (interpolated
  between d.f. 60 and d.f. 120).

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• 6. Interpret
• In this case, the calculated value is greater
  than the critical value (1.786 gt 1.66),
• so we reject the null hypothesis and conclude
  that the average accounts receivable outstanding
  has increased.

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CHI SQUARE TESTS

• The symbol is
• This may be used in non-parametric and parametric
  tests
• Effective for categorical data
• persons, events, or objects
• grouped in two or more nominal categories
• such as "yes-no," "favour-undecidedagainst," or
  class "A, B, C, orD

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CHI SQUARE TESTS

• Using this technique, we test for significant
  differences between the observed distribution of
  data among categories and the expected
  distribution based on the null hypothesis.
• Chi-square is useful in cases of one-sample
  analysis, two independent samples, or k
  independent samples.
• It must be calculated with actual counts rather
  than percentages.

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CHI SQUARE TESTS

• In the one-sample case, we establish a null
  hypothesis based on the expected frequency of
  objects in each category
• Then the deviations of the actual frequencies in
  each category are compared with the hypothesised
  frequencies.
• The greater the difference between them, the less
  is the probability that these differences can be
  attributed to chance.

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• The value of chi-square is the measure that
  expresses the extent of this difference.
• The larger the divergence, the larger is the
  chi-square value.

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• There is a different distribution for X2 for each
  number of degrees of freedom (d.f.), defined as
  (k - I) or
• the number of categories in the classification
  minus 1.
• df k - 1

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• Lets take an example
• We have interviewed 200 students and learned of
  their intentions to a eating club. We would like
  to analyse the results by living arrangement
  (type and location of student housing and eating
  arrangements). The 200 responses are classified
  into the four categories shown in the
  accompanying table. Do these variations indicate
  there is a significant difference among these
  students, or are these sampling variations only?

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• A 6 step procedure is needed
• 1. Null hypothesis
• 2. Statistical test
• 3. Significance level
• 4. Calculated value
• 5. Critical test value
• 6. Interpret

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• 1. Null hypothesis
• Ho Oi Ei
• The proportion in the population who intend to
  join the club is independent of living
  arrangement.
• In HA Oi Ei
• the proportion in the population who intend to
  join the club is dependent on living arrangement.

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• 2. Statistical test
• Use the one-sample X2 to compare the observed
  distribution to a hypothesized distribution.
• The X2 test is used because the responses are
  classified into nominal categories and there are
  sufficient observations
• 3. Significance level
• Let alpha .05.

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4. Calculated value
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• 5. Critical test value
• Enter the table of critical values of X2 (see
  table), with 3 d.f., and secure a value of 7.82
  for a .05.
• 6. Interpret
• The calculated value is greater than the critical
  value, so the null hypothesis is rejected.

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ANALYSIS OF VARIANCE (ANOVA)

• ANOVA is an extremely useful technique concerning
  researches in the fields of economics, biology,
  education, psychology, sociology,
  business/industry and in researches of several
  other disciplines.
• This technique is used when multiple sample cases
  are involved.

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ANALYSIS OF VARIANCE

• The significance of the difference between the
  means of two samples can be judged through either
  z-test or the t-test, but the difficulty arises
  when we happen to examine the significance of the
  difference amongst more than two sample means at
  the same time.

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ANALYSIS OF VARIANCE

• The ANOV A technique enables us to perform this
  simultaneous test and as such is considered to be
  an important tool of analysis in the hands of a
  researcher.
• Using this technique, one can draw inferences
  about whether the samples have been drawn from
  populations having the same mean.

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ANALYSIS OF VARIANCE

• ANOVA technique is important in the context of
  all those situations where we want to compare
  more than two populations such as in comparing
  the yield of crop from several varieties of
  seeds, the gasoline mileage of four automobiles,
  the smoking habits of five groups of university
  students and so on.

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ANALYSIS OF VARIANCE

• In such circumstances one generally does not want
  to consider all possible combinations of two
  populations at a time for that would require a
  great number of tests before we would be able to
  arrive at a decision.
• This would also consume lot of time and money,
  and even then certain relationships may be left
  unidentified (particularly the interaction
  effects).

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• Therefore, one quite often utilises the ANOVA
  technique and through it investigates the
  differences among the means of all the
  populations simultaneously.

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THE BASIC PRINCIPLE OF ANOVA

• The basic principle of ANOVA is to test for
  differences among the means of the populations by
  examining the amount of variation within each of
  these samples, relative to the amount of
  variation between the samples.

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• In terms of variation within the given
  population, it is assumed that the values of (X)
  differ from the mean of this population only
  because of random effects
• i.e., there are influences on (Xij) which are
  unexplainable, whereas in examining differences
  between populations we assume that the difference
  between the mean of the jth population and the
  grand mean is attributable to what is called a
  'specific factor' or what is technically
  described as treatment effect.

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• Thus while using ANOVA, we assume that each of
  the samples is drawn from a normal population and
  that each of these populations has the same
  variance. We also assume that all factors other
  than the one or more being tested are effectively
  controlled.
• This means that we assume the absence of many
  factors that might affect our conclusions
  concerning the factor(s) to be studied.

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• In short, we have to make two estimates of
  population variance viz., one based on between
  samples variance and the other based on within
  samples variance.
• Then the said two estimates of population
  variance are compared with F-test.

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The F Distribution

• The probability distribution used in this chapter
  is the F distribution.
• It was named to honor Sir Ronald Fisher, one of
  the founders of modern-day statistics.
• This probability distribution is used as the
  distribution of the test statistic for several
  situations.
• It is used to test whether two samples are from
  populations having equal variances, and it is
  also applied when we want to compare several
  population means simultaneously.

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The F Distribution

• Note the simultaneous comparison of several
  population means is called analysis of variance
  (ANOVA).
• In both of these situations, the populations must
  follow a normal distribution, and the data must
  be at least interval-scale.
• What are the characteristics of the F
  distribution?

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Characteristics of F Distribution

• There is a "family" of F distributions
• A particular member of the family is determined
  by two parameters
• the degrees of freedom in the numerator
• the degrees of freedom in the denominator.
• The shape of the distribution is illustrated on
  the next slide
• There is one F distribution for the combination
  of 29 degrees of freedom in the numerator and 28
  degrees of freedom in the denominator.

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Characteristics of F Distribution

• There is another F distribution for 19 degrees in
  the numerator and 6 degrees of freedom in the
  denominator.
• Note that the shape of the curves change as the
  degrees of freedom change.

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Characteristics of F Distribution

• 2. The F distribution is continuous
• It can assume an infinite number of values
  between zero and positive infinity
• 3. The F distribution cannot be negative
• The smallest value F can assume is 0.
• 4. It is positively skewed
• The long tail of the distribution is to the
  right-hand side.
• As the number of degrees of freedom increases in
  both the numerator and denominator the
  distribution approaches a normal distribution.

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Characteristics of F Distribution

• 5. It is asymptotic
• As the values of X increase, the F curve
  approaches the X-axis but never touches it.
• This is similar to the behavior of the normal
  distribution

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Example

• Lets look at the SPSS results.

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