Str t-test1
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This document describes the steps for conducting an independent samples t-test. The t-test is used to compare the means of two independent groups on a continuous dependent variable. It tests whether the means of the two groups are statistically significantly different from each other. The steps include: 1) stating the null and alternative hypotheses, 2) setting the significance level, 3) calculating the t-value, 4) finding the critical t-value, and 5) making a conclusion about whether to reject the null hypothesis based on the t-values. An example compares math test scores of male and female college students to determine if gender significantly impacts scores.
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Descriptive Statistics Formula Sheet
Sample Population
Characteristic statistic Parameter
raw scores x, y, . . . . . X, Y, . . . . .
mean (central tendency) M =
∑ x
n
μ =
∑ X
N
range (interval/ratio data) highest minus lowest value highest minus lowest value
deviation (distance from mean) Deviation = (x − M ) Deviation = (X − μ )
average deviation (average
distance from mean)
∑(x − M )
n
= 0
∑(X − μ )
N
sum of the squares (SS)
(computational formula) SS = ∑ x
2 −
(∑ x)2
n
SS = ∑ X2 −
(∑ X)2
N
variance ( average deviation2 or
standard deviation
2
)
(computational formula)
s2 =
∑ x2 −
(∑ x)2
n
n − 1
=
SS
df
σ2 =
∑ X2 −
(∑ X)2
N
N
standard deviation (average
deviation or distance from mean)
(computational formula) s =
√∑ x
2 −
(∑ x)2
n
n − 1
σ =
√∑ X
2 −
(∑ X)2
N
N
Z scores (standard scores)
mean = 0
standard deviation = ± 1.0
Z =
x − M
s
=
deviation
stand. dev.
X = M + Zs
Z =
X − μ
σ
X = μ + Zσ
Area Under the Normal Curve -1s to +1s = 68.3%
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-3s to +3s = 99.7%
Using Z Score Table for Normal Distribution
(Note: see graph and table in A-23)
for percentiles (proportion or %) below X
for positive Z scores – use body column
for negative Z scores – use tail column
for proportions or percentage above X
for positive Z scores – use tail column
for negative Z scores – use body column
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1. Convert each X to Z score
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Regression lines
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points; used for predictions
only) formula uses raw
scores
b = slope
a = y-intercept
y = bx + a
(plug in x
to predict y)
b =
∑ xy −
(∑ x)(∑ y)
n
∑ x2 −
(∑ x)2
n
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Pearson Correlation Coefficient
(used to measure relationship;
uses Z scores)
r =
∑ xy−
(∑ x)(∑ y)
n
√(∑ x2−
(∑ x)2
n
)(∑ y2−
(∑ y)2
n
)
r =
degree x & 𝑦 𝑣𝑎𝑟𝑦 𝑡𝑜𝑔𝑒𝑡ℎ𝑒𝑟
degree x & 𝑦 𝑣𝑎𝑟𝑦 𝑠𝑒𝑝𝑎𝑟𝑎𝑡𝑒𝑙𝑦
r
2
= estimate or % of accuracy of predictions
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level?
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b. What is a type II error? How does it happen?
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Str t-test1
- 1. T-test for Two Independent Samples Group 1: Emman, Misola, Peralta, Sulam, Dumagpi, Tamayo • This is probably the most widely used statistical test of all time, and certainly the most widely known. It is simple, straightforward, easy to use, and adaptable to a broad range of situations. • It is a method of comparing two independent populations by evaluating the mean difference. The measurement is usually expressed in either interval or ratio. • A third variable (a dependent variable) is measured using continuous scale of measurement. This variable will be used as a reference for comparison of the two independent variables. • The parameters are unknown. Application: • It is best utilized in researches trying to answer the question: “If we examine two different levels of one variable, will we find them to be associated with different levels of the other?” • Assumptions when doing the test: – The two groups are independent of each other. – The dependent variable is normally distributed. – The two groups have approximately equal variance on the dependent variable. Steps • Step 1: State the Null Hypothesis. – The null hypothesis is the main antithesis of the research. It is the hypothesis of non-significance. It is expressed as Ho: µ1 = µ2, µ1 ≤ µ2 or µ1 ≥ µ2. Basically the research aims to disprove this assumption – by evaluating if there is any significance in the data results. • Step 2: State the alternative hypothesis. – The alternative hypothesis, denoted as HA , is the hypothesis of significance – that there is significance in the mean differences of the two populations. It is expressed as HA: µ1 ≠ µ2, µ1 < µ2 or µ1 > µ2. • Step 3: Set the alpha level. α = 0.05 is the most commonly used level of significance. – This can be expressed in decimals such as 0.1, 0.5, etc. This defines the range at which the range of Ho rejection under the standard curve may be. • Step 4: Do the calculations to get tcalculated – First compute for the means of samples 1 and 2. – Then Compute for standard error of the differences of the two means. – Lastly, compute for the t (t = (sample mean1 – sample mean 2)/standard error) t = x̄1 - x̄2 f------------------> Difference of the sample means sqrt(s1 2 /n1 + s2 2 /n2) -------------------> Standard Error • Step 5: Find the critical or tabular t - Get the degree of freedom (df) of the test. - Then, in the Table of Critical Value of t (refer to the research manual), get the tcritical. In the y axis is the degrees of freedom and in the x axis is the alpha. Consider whether the study is a one-tailed or a two- tailed test. Where: x̄ = sample mean s2 = standard deviation of sample (denoted as xόn-1 in common calculators) n = number of samples (sample size) df = n1 + n2 - # of sample groups
- 2. • Step 6: Decide whether the t value is within the range of Ho or not (CONCLUDE) – For one tailed test, if the tcalculated is to the left of the tcritical (if µ1 < µ2) or to the right of the tcritical (if µ1 > µ2), then the HA is correct. If not, then Ho is correct. – For two tailed test, if the tcalculated is within the range of +tcritical, then Ho is true. If not within the range, then the Ha is true. Example A study was conducted to determine if gender plays a significant role in math test performances of college students. Six students of each gender were given a surprise 30 – item quiz. The result follows: 1. State null hypothesis There is no significant difference between the performances of the two genders in the Math test. 2. State alternative hypothesis There is a significant difference between the performances of the two genders in the Math test. 3. State the α level = 0.05. 4. Get the tcalculated t = x̄1 - x̄2 f sqrt(s1 2 /n1 + s2 2 /n2) * x̄1 = 21.5 (average for the male scores) * x̄2 = 19.5 (average for the female scores) * s1 2 = standard deviation of males squared *s2 2 = standard deviation of females squared *n1 = number of male samples *n2 = number of female samples !! Get the standard deviation in the calculator by pressing xόn-1 after inputting the data for each sample group. t = 21.5 – 19.5 f sqrt(7.232 /6 + 2.592 /6) = 2 / sqrt (8.71 + 1.12 ) = +0.6379 or +0.64 5. Find tcritical Get the tdf = n1 + n2 - # sample groups = 6 + 6 - 2 = 10 So… 10 vs. 0.05 = (+)2.28 two – tailed so plus-minus 6. Conclude: -2.28 < 0.64 < +2.28 - T is within the range of t+critical: Ho is TRUE: - There is no significant difference between the performances of the two genders in the Math test. Sources • shoffma5(2008). T Test For Two Independent Samples. Retrieved Nov. 7, 2010 from http://www.slideshare.net/shoffma5/t- test-for-two-independent-samples Scores Males Females 22 20 29 21 16 23 27 16 10 17 25 20
- 3. • Lowry R. (1999-2010). Chapter 11. t-Test for the Significance of the Difference between the Means of Two Independent Samples. Retrieved Nov. 7, 2010 from http://faculty.vassar.edu/lowry/ch11pt1.html • MacFarland T.W. (1998). Student's t-Test for Independent Samples. Retrieved Nov. 7, 2010 from http://www.nyx.net/~tmacfarl/STAT_TUT/studen_t.ssi