The following data was collected from a group of students who are taking a statistics course:
Subject #
|
gender
|
breakfast
|
Sleep Hrs
|
Study Hrs
|
Quiz
|
Faculty
|
1
|
f
|
n
|
7
|
6
|
8
|
arts
|
2
|
f
|
y
|
7
|
5
|
12
|
med
|
3
|
f
|
n
|
9
|
5
|
9
|
arts
|
4
|
f
|
n
|
6.5
|
8
|
7
|
arts
|
5
|
f
|
n
|
8
|
8
|
12
|
arts
|
6
|
m
|
y
|
8
|
8
|
8
|
arts
|
7
|
f
|
y
|
6.5
|
6
|
10
|
sci
|
8
|
f
|
n
|
6.5
|
6
|
3
|
sci
|
9
|
f
|
n
|
7
|
4.5
|
3
|
arts
|
10
|
f
|
y
|
7.5
|
8.5
|
9
|
sci
|
11
|
m
|
n
|
7
|
4
|
8
|
arts
|
12
|
m
|
n
|
9
|
5
|
9
|
arts
|
13
|
m
|
n
|
6.5
|
8.5
|
8
|
arts
|
14
|
m
|
y
|
7
|
2
|
9
|
sci
|
Specify which test would be used for each of the following hypotheses.
1) Males and females differ in whether or not they regularly eat breakfast.
2) The breakdown of students into faculty in PSYC 2020 is not uniform.
3) Students invest more hours in sleep each night than they do in stats all week.
4) Study hours depend on quiz scores.
5) Students in Arts and Science perform differently on the quiz.
Conduct the tests for #3 and #5 and write a short report for each.
For #1, #2 and #4, state only H0and H1and df and critical value, but no calculation is necessary.
Microsoft Word - Assignment1_Draft.docx Statistical Methods I and II PSYC 2020 B A S S I G N M E N T # 1 D u e M o n d a y , N o v . 2 5 , 2 0 1 9 , i n c l a s s ALL QUESTIONS should be ATTEMPTED and NEATLY ORGANIZED. SHOW YOUR WORK. The following data was collected from a group of students who are taking a statistics course: Subject # gender breakfast Sleep Hrs Study Hrs Quiz Faculty 1 f n 7 6 8 arts 2 f y 7 5 12 med 3 f n 9 5 9 arts 4 f n 6.5 8 7 arts 5 f n 8 8 12 arts 6 m y 8 8 8 arts 7 f y 6.5 6 10 sci 8 f n 6.5 6 3 sci 9 f n 7 4.5 3 arts 10 f y 7.5 8.5 9 sci 11 m n 7 4 8 arts 12 m n 9 5 9 arts 13 m n 6.5 8.5 8 arts 14 m y 7 2 9 sci Specify which test would be used for each of the following hypotheses. 1) Males and females differ in whether or not they regularly eat breakfast. 2) The breakdown of students into faculty in PSYC 2020 is not uniform. 3) Students invest more hours in sleep each night than they do in stats all week. 4) Study hours depend on quiz scores. 5) Students in Arts and Science perform differently on the quiz. Conduct the tests for #3 and #5 and write a short report for each. For #1, #2 and #4, state only H0 and H1 and df and critical value, but no calculation is necessary. Instructor: Ji Yeh Choi (
[email protected]) Unless otherwise noted, all material posted for this course are copyright of the instructor, and cannot be reused or reposted without the instructor’s written permission. Statistical methods 1 York University Psychology Department PSYC 2020 - Statistical Methods I and II Topic 7: Introduction to the T-statistic Outline for today 2 • Review of Steps for Hypothesis Testing • Tests of Means Statistical methods Statistical methods 3 n Step 1: Set up a hypothesis n Null hypothesis (H0): n No effect (“People will be equally fat regardless of how many hamburgers they eat.”) n This is what we test statistically. n Alternative hypothesis (H1): n Some effect (“People eating more hamburgers will be fatter than those eating less hamburgers.”) n research/experimental hypothesis. Steps for Hypothesis Testing Statistical methods 4 n Step 2: Choose a (significance level) n Decide the area consisting of extreme scores which are unlikely to occur if the null hypothesis is true. n Conventionally, a = .05 (or .01). n The cutoff sample score for a is called the critical value. Steps for Hypothesis Testing[ If; D= .05 O Aioli Statistical methods 5 n Step 3: Examine empirical data and compute the appropriate test statistic. Test statistics n Means (one) z (if s is known) t (if s is unknown) n Means (two) t n Means (more than two) ANOVA Steps for Hypothesis Testing or¥8s . In Statistical methods 6 n Step 4: Make the decision whether to ‘reject’ or ‘not reject’ the null hypothesis. n Compare the calculated value of your test statistic to the (tabled) critical value for a. n If your value is greater than or equal to that the critical value, reject H0. Otherwise, retain H0. n A decision to reject H0 implies an acceptance of H1. Steps for Hypothesis Testing ①ot> arena - or÷HI⇒snsea Statistical methods 7 n Step 4: Make the decision whether to ‘reject’ or ‘not reject’ the null hypothesis. n Alternatively, look at the significance level (p-value) of your test statistic value. n If p-value ≤ .05, reject H0. Otherwise, retain H0. n If H0 is rejected, you may conclude that there is a statistically significant effect in the population. Steps for Hypothesis Testing P > . 05 . fail to reject Ho .⑤ tejea Ho . •*ai7* Hypothesis Testing Statistical methods 8 § Hypothesis testing of a single mean § z-test § One-sample t-test (or called single sample t-test: comparing one SAMPLE mean against a POPULATION mean) § Hypothesis testing of two means § Independent-samples t-test (between-subjects design) § Dependent-samples t-test (within-subjects design) Mtoadffenences. I 9¥77:* sentence⑧④: . ..÷÷i÷÷÷ . ¥:3 a:* . W=¥EeEf¥ps . - ← Statistical methods 9 Hypothesis Testing of a Single Mean – z-test sample = 110X n H0 : μ = 100 n H1 : μ ¹ 100 IQ 25 students from PSYC 2020 Canadian Population μ = ? σ = 15 n - Hi . Statistical methods 10 Hypothesis Testing of a Single Mean – z-test n Purpose: n to test whether a sample mean ( ) differs from a population mean (μ). n Prior Requirements/Assumptions: n The population is normally distributed. n The population standard deviation (σ) must be known. n The sample must be a simple random sample of the population (independence of observations). X * Statistical methods 11 Hypothesis Testing of a Single Mean – z-test n Technically, the z-test for a single mean is very similar to calculating the z score of your sample mean. Statistical methods 12 n We can convert our sample score (X) to a standard score (z) which follows the standard normal distribution. s µ- = Xz μ = 0 & σ = 1 z ~ N(0,1) Deviation of a sample score from pop. mean Limits the range of the deviation Z (Standard) score RAW data . - X . . g-mega:tf O Statistical methods 13 Hypothesis Testing of a Single Mean – z-test n Apply the z-statistic to your sample mean ( ) with µ and σ for the sample mean. M MXz s µ- = σ μXz -= Z-statistic for sample meanZ-statistic for sample score µM = µ (population mean) σM = σ ! (N = sample size) X " iii.heooo . ' a En O Try-- Fx Statistical methods 14 N(0, 1) Z = 1.96Z = -1.96 .025 (2.5%) Critical value for α = .05 Standard Normal Distribution Ho &> 951 Sym non . - eefeonal . after - finding -2 -statistic. VS . I 1.96 7--1.96 Iz - staff > III. 961 ⇒ reject Ho . • •• Statistical methods 15 Canadian Population sampleμ = 100 σ = 15 = 110X n Ho : μ = 100 n H1 : μ ¹ 100 n µM = µ = 100 n σM = σ " = #$ %$ = 3 n z = ##&'#&&( = 3.33N = 25 Q: What would be your decision then? IQ Z-test: Example Ho: NEKO He : M > Coo . zI④q="o7÷o÷ -15=3.33 8O + Hoo - 7100 ' 2- =3 . ?3 > Zora . ⇒ reject to . Statistical methods 16 Z-test: Example n The null hypothesis may be rejected because the z- statistic value (3.33) is greater than 1.96. Instead, the alternative hypothesis may be accepted. n This indicates that the mean IQ score statistically significantly differs from the mean IQ score of the Canadian population at the .05 level (Z = 3.33, p < .05).="" -="" -="" -="" "="" reject="" ho="" '="" '="" p=""> . 05 ⇐ Limitations of Z-test Statistical methods 17 n Knowing the true value of the standard deviation (σ) of a population is unrealistic. n Alternative? - t-test n σ is unknown Statistical methods 18 Hypothesis Testing of a Single Mean – t-test n Purpose: n to test whether a sample mean ( ) differs from a population mean (μ). n Prior Requirements/Assumptions: n The population is normally distributed. n The sample must be a simple random sample of the population (independence of observations). X - - Statistical methods 19 Canadian Population X sampleμ = ? σ = ? n Ho : μ = 100 n H1 : μ ¹ 100 IQ M MXz s µ- = = 110 σM = ? N μ N = 25 s = 14 Sample standard deviation Hypothesis Testing of a Single Mean Z -- Est t . out . = or z=.⑤ ① samples .de ) o ②*€=XO . P Sm or Sx --14nF . Statistical methods 20 Canadian Population X sampleμ = ? σ = ? N = 25 s = 14 n Ho : μ = 100 n H1 : μ ¹ 100 IQ M M s Xt µ-= = 110 sM = s N μ t statistic Hypothesis Testing of a Single Mean – t-test O . .÷÷: . = 142 Fuse . - To 110 - I 00 t - - - it RE Statistical methods 21 n Varies in shape according to the degrees of freedom (DF). n The t distribution approaches the standard normal distribution as DF becomes large. n The approximation is quite good for DF > 30. The t-distribution ④=H ensteadfast . we thou - -④→ a ⑧ as sma Statistical methods 22 t0 t (df = 5) t (df = 13) t-distributions are bell- shaped and symmetric, but have ‘fatter’ tails than the normal Standard Normal (t with df = ¥) Note: t z as N increases from “Statistics for Managers” Using Microsoft® Excel 4th Edition, Prentice-Hall 2004 The t-distribution 1107mi Statistical methods 23 Hypothesis Testing of a Single Mean – t-test n Calculate the value of t-statistic (or simply t-value) for your sample mean ( ) with µM and sM. n Degrees of freedom for a single sample t-test = N - 1 n Look into how extreme your t-value is. n Look at the table of t values and find the critical value for the significance level of .05, given DF. n For example, when DF = 24, critical value = 2.064 at .05 level X ° # Eof means . potpulateo Pineau . - - - • twang:Eai¥ * IDOH to ) Statistical methods 24 Hypothesis Testing of a Single Mean – t-test n If your t-value in absolute value is greater than or equal to the critical value, you may reject the null hypothesis at the significance level of .05. n Alternatively, look at the p-value/ significance