SlideServe
Télécharger Obtenir Pro
Menu
Accueil Pro
Préférences
ApparenceLight Aide et commentaires
© SlideServe · v2026

COMPARING MEANS: INDEPENDENT SAMPLES

marcin avatar
marcin
10 diapositives · 356 Views · Oct 10, 2014

COMPARING MEANS: INDEPENDENT SAMPLES. 1 ST sample: x1, x2, …, xm from population with mean μ x; 2 nd sample : y1, y2, …, yn from population with mean μ y; GOAL: Determine if μ x = μ y based on the two samples.

COMPARING MEANS: INDEPENDENT SAMPLES - slide 1
01 / 10COMPARING MEANS: INDEPENDENT SAMPLES
1 / 17

Télécharger la présentation

COMPARING MEANS: INDEPENDENT SAMPLES
Une image / un lien ci-dessous est fourni (tel quel) pour télécharger la présentation

Politique de téléchargement : le contenu du site vous est fourni TEL QUEL pour votre information et votre usage personnel ; il ne peut être vendu, concédé sous licence ni partagé sur d’autres sites sans l’accord de son auteur.Le contenu vous est fourni TEL QUEL, pour votre information et votre usage personnel uniquement. Téléchargez la présentation en cliquant sur ce lien. Lors du téléchargement, si vous ne parvenez pas à télécharger une présentation, l’éditeur a peut-être supprimé le fichier de son serveur.Pendant le téléchargement, si vous ne parvenez pas à obtenir une présentation, le fichier a peut-être été supprimé par l’éditeur.

F I N

Plus de contenu associé

Transcription de la présentation

  1. COMPARING MEANS: INDEPENDENT SAMPLES 1ST sample: x1, x2, …, xm from population with mean μx; 2nd sample: y1, y2, …, yn from population with mean μy; GOAL: Determine if μx = μy based on the two samples. Test Ho: μx = μy vs Ha: μx ≠ μy or Ha: μx > μy or Ha: μx < μy Procedure depends on what can we assume about variability of the populations: σx and σy. CASE1. σx and σy are known. CASE2. σx and σy are not known, but may be assumed equal σx=σy CASE3. σx and σy are not known, and can not be assumed equal. Test statistics are developed for each of the 3 cases.

  2. COMPARING MEANS: INDEPENDENT SAMPLESCASE 1: σx and σy known Test on significance level α. STEP1.Ho: μx = μy vs Ha: μx ≠ μy or Ha: μx > μy STEP 2. Test statistic: Under the Ho, the test statistic has standard normal distribution. STEP 3. Critical value? For one-sided test zα, for two-sided zα/2 . STEP 4. DECISION-critical/rejection region(s) depends on Ha. Ha: μ ≠ μo Reject Ho if |z|> zα/2; Ha: μ > μo Reject Ho if z > zα; Ha: μ < μo Reject Ho if z < - zα. STEP 5. Answer the question in the problem.

  3. COMPARING MEANS: INDEPENDENT SAMPLESCASE 2: σx and σy not known, but assumed equal. STEP 2. Test statistic: where is a pooled estimate of the common variance Under the Ho, the test statistic has t distribution with df = m+n-2. STEP 3. Critical value? One-sided test tα, two-sided tα/2 . STEP 4. DECISION-critical/rejection region(s) depends on Ha. Ha: μ ≠ μo Reject Ho if |t|> tα/2; Ha: μ > μo Reject Ho if t > tα; Ha: μ < μo Reject Ho if t < - tα.

  4. COMPARING MEANS: INDEPENDENT SAMPLESCASE 3: σx and σy not known, and may not be assumed equal. STEP 2. Test statistic: Under Ho, the degrees of freedom for the t distribution may be approximated by df=min(m-1, n-1). STEP 3. Critical value? One-sided test tα, two-sided tα/2 . STEP 4. DECISION-critical/rejection region(s) depends on Ha. Ha: μ ≠ μo Reject Ho if |t|> tα/2; Ha: μ > μo Reject Ho if t > tα; Ha: μ < μo Reject Ho if t < - tα.

  5. EXAMPLE1 A medication for blood pressure was administered to a group of 13 randomly selected patients with elevated blood pressure while a group of 15 was given a placebo. At the end of 3 months, the following data was obtained on their Systolic Blood Pressure. Control group, x: n=15, sample mean = 180, s=50 Treated group, y: m=13, sample mean =150, s=30. Test if the treatment has been effective. Assume the variances are the same in both groups and use α=0.01. Soln. Let μx= mean blood pressure for the control group; μy= mean blood pressure for the treatment group. Then, n=15, = 180, sx=50, m=13, =150, sy =30. Assumed equality of variances/st.dev. σx=σy

  6. EXAMPLE1 contd. STEP1.Ho: μx = μy (medicine not effective) vs Ha: μx > μy (med. effective) STEP 2. Pooled variance: Standard deviation Test statistic: STEP 3. Critical value=t0.01=2.479, df=26. STEP 4. t=1.8863 not > 2.479, do not reject Ho. STEP 5. Not enough evidence to conclude that the medicine is effective.

  7. (1-α)100% CONFIDENCE INTERVAL FOR (μx- μy) CASE 1: σx and σy known CASE 2: σx and σy not known, but assumed equal. Use t distribution with df=m+n-2. CASE 3: σx and σy not known, and may not be assumed equal. Use t distribution with df=min(m-1, n-1).

  8. EXAMPLE1 contd. Construct a 95% CI for the difference in the means of blood pressures for the two groups (μx - μy). Soln. We already know n=15, = 180, sx=50, m=13, =150, sy =30, sp=41.97. CASE 2. 95% CI, so α=0.05, so α/2=0.025, t(26)0.025 = 2.056. 95% CI is: NOTE: The interval contains zero. Intuitively, that conforms our decision that there is no difference in means between the medicine and the placebo.

  9. MINITAB EXERCISE Is there a significant difference between test scores on the 1st and 2nd test? Use data in example176.xls (see class web site).

  10. MINITAB EXAMPLE contd. Two Sample T-Test and Confidence Interval Two sample T for exm1 vs exm2 N Mean StDev SE Mean exm1 30 77.2 10.5 1.9 exm2 30 69.0 21.9 4.0 95% CI for mu exm1 - mu exm2: ( -0.8, 17.1) T-Test mu exm1 = mu exm2 (vs not =): T = 1.85 P = 0.072 DF = 41