Blood pressure of 60 year old women with glaucoma
The confidence interval for the population mean is given by:
N
200
Mean
140
Standard Deviation
25
SE Mean
1.77
Confidence interval for population mean is (136.51, 143.49)
There is 95% confidence that true mean will be between 136.51 and 143.49
The confidence interval for the sample mean is given by:
N
100
Mean
140
Standard Deviation
25
SE Mean
2.5
The 95% confidence interval for sample mean is (135.04, 144.96)
There is 95% confidence that true mean will be between 135.04 and 144.96
The interval gets wider when the sample size is decreased by 100 from (136.51, 143.49) to (135.04, 144.96).
Reason: Because reducing the sample size (n) increases the standard error of the mean from 1.77 to 2.5 hence the confidence interval gets wider.
Paired t-test/Confidence interval for matched samples
Then the 95% confidence interval for the mean difference is given by:
Difference
Sample size
107
Mean
2.1
Standard Deviation
3.1
SE Mean
0.3
The 95% confidence interval for mean difference is (1.506, 2.694)
There is 95% confidence that true mean is located in the interval of (1.506, 2.694), it means that PTCA is effective in increasing exercise duration. The problem of constructing confidence interval for the difference of the means from two populations with unknown variances is difficult.
The 95% confidence for the mean difference is given by:
Vitamin E
Placebo
Sample size
81
90
Mean
27
24
Standard Deviation
6.9
6.20
SE Mean
0.77
0.65
The 95% confidence interval for the sample difference is (1.02, 4.99)
Two sample t test
The problem of unknown variances of the two groups.
There is 95% certainty that true mean is located between 1.02 and 4.99, thus the conclusion is Vitamin E increases cognitive ability among elderly women.
The test hypothesis is
Ho: There is a difference in cooling constants between freshly killed and reheated corpse. Vs Ha: There is no difference in cooling constants between freshly killed and reheated corpse.
Freshly Killed
Reheated
Difference
Sample size
19
19
19
Mean
417.9
376.1
41.8
Standard Deviation
60
74.7
91.6
SE mean
13.8
17.1
21.0
There 95% CI for mean difference is (-86,2.3)
T-Value = -1.99
P-Value = 0.062
Null hypothesis is not rejected, because P- value 0.062 is greater than 0.05 and therefore not significant. Also since the confidence interval contains 0, its not significant. This means there is no difference in cooling rates when rats were killed and reheated.
Parameters
Difference
Sample size (n)
120
Mean
0.130
Standard Error (SED)
0.053
The 95% credible interval for the mean difference is between the interval (0.026, 0.234)
There is 95 % certainty that the difference is between the value of 0.026 and 0.234. Yes, there is the difference.
U.S army recruits in Iraq
Yes, the measurements are still affected by sampling error. When the size of the sample is reduced by random sampling, it increases the sample error.
The recruits are chosen at random hence will reduce biasness, hence the recruit will have better chance of representing the whole population.
The above study is an observational study, because the researcher has no influence on how the fishes fall into groups. Therefore the study is purely observational.
The two variables associated with this study are: subspecies (group) of guppy in Venezuela and most sensitivity wavelength of light.
The explanatory variable in this study is group of guppy in Venezuela and the response variable is wavelength of light.
(In log 10 units)
N
39
39
Mean
5.50
155.50
Standard Deviation
0.2555
5.36
The test hypothesis associated with chi-square test is:
Ho: Treatments and response are independent
Versus
Ha: Treatments and response are dependent
The chi-square test is given by:
Treatment
Favorable
Unfavorable
Placebo
28.90
35.10
Test drug
27.10
32.90
Pearson Chi-Square = 21.709, DF = 1, P-Value = 0.000
Likelihood Ratio Chi-Square = 22.377, DF = 1, P-Value = 0.000
Reject the null hypothesis that treatment and response are independent, because the p value is less than 0.05. Hence. treatments and response are dependent.
We need to test the hypothesis
Ho : There is no change of testorone levels vs. Ha : There is a change of testorone values.
After
Before
Difference
Sample size
4
4
4
Mean
88.50
91.50
-3
Standard Deviation
91.50
7.33
2.58
SE mean
2.72
3.66
1.29
95% CI for mean difference: (-7.11, 1.11)
T-Test of mean difference = 0 (vs ≠ 0): T-Value = -2.32 P-Value = 0.103
We don't reject the null hypothesis that "There is no change of testorone levels" because the P. Value =0.103 is more than the significant value(0.05). Also because the confidence interval contains 0 hence its not significant. Hence, There is no change of testorone levels taken before and after watching a game.
Mean
2.26
Median
1.986
Standard Deviation
1.787
Coefficient of variation
79.07%
One tailed test
Two tailed test
t(0.01, 10)
2.764
±3.169
t0.99, 10
-2.764
±0.01285
t0.025, 7
2.365
±2.841
t0.975, 7
-2.365
±0.03248
t0.005, 23
2.807
±3.104
t0.995, 23
-2.807
±0.006335
N
12
Mean
16.69
Standard Deviation
0.704
SE Mean
0.203
The confidence interval for the sample mean is (16.244, 17.139)
There is 95% confidence that the true mean will be between the intervals (16.244, 17.139)
The formulated null hypothesis is:
Ho: Time taken is the same for both calculators Versus Ha: Time taken is not the same between the calculators.
Sample
Calculator A
Calculator B
Sample size(n)
12
12
Mean
24.67
23.17
Standard Deviation
4.89
4.17
SE Mean
1.4
1.2
Difference = μ (B) - μ (A)
Estimate for difference: -1.50
95% CI for difference: (-5.36, 2.36)
T-Test of difference = 0 (vs ≠): T-Value = -0.81 P-Value = 0.428 DF = 21
Don't reject the null hypothesis that time taken is the same for both calculators because P-value is more than significant (0.05). Hence time taken is the same for both calculators to process result.
The test hypothesis associated test is:
Ho: There is no association between hair and color in human beings.
Versus
Ha: There is association between hair and color in human beings.
x
Dark hair
Light hair
Total
Brown eyes
782
241
1023
Blue eyes
234
60
294
Total
1016
301
1317
Dark hair
Light hair
Brown Eyes
789.2
233.8
Blue Eyes
226.8
67.2
Pearson Chi-Square = 1.285, DF = 1, P-Value = 0.257
Likelihood Ratio Chi-Square = 1.310, DF = 1, P-Value = 0.252
Dont reject the null hypothesis that There is no association between hair and color in human beings because the p-value is more than 0.05. Hence, There is no association between hair color and human beings.
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