Standard Deviation of The Population Essay Assignment

Standard Deviation of The Population Essay Assignment

Question 1

The sample mean that will probably give out a more accurate estimate of the mean of population will be the one with a smaller standard error of the mean (SEM). The SEM formula is:

SEM = SD / sqrt(n)

where SD is the standard deviation of the population, and n is the sample size.

For the sample from Population A:

SEM = 3.2 / sqrt(30) = 0.585

For the sample from Population B:

SEM = 3.2 / sqrt(55) = 0.430

Compared to the sample from Population A, the SEM from Population B is smaller. Hence, compared to the sample mean from Population A, the sample mean from Population B is more likely to produce an estimate of its population mean that is more correct. Standard Deviation of The Population Essay Assignment

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The sample from Population B has a greater sample size than the sample from Population A, and the SEM declines as sample size grows.

Question 2

We must utilize the t-distribution table to determine the confidence intervals for a 95% and 99% level of confidence.

We use a two-tailed t-test with n – 1 = 39 degrees of freedom (df) and a t-value of 2.024 (from the t-distribution table with 39 degrees of freedom and a probability of 0.025 in each tail) for a 95% confidence interval:

Margin of error (ME) = t-value * standard error of the mean (SEM)

SEM = standard deviation (SD) / sqrt(sample size) = 1.1 / sqrt(40) = 0.174

ME = 2.024 * 0.174 = 0.352

Therefore, the 95% confidence interval is:

CI = sample mean ± ME = 7.1 ± 0.352 = (6.748, 7.452)

We employ a two-tailed t-test with degrees of freedom (df) of n – 1 = 39 and a t-value of 2.704 to determine the confidence interval at 99%. (from the t-distribution table with 39 degrees of freedom and a probability of 0.005 in each tail)

Margin of error (ME) = t-value * standard error of the mean (SEM)

SEM = standard deviation (SD) / sqrt(sample size) = 1.1 / sqrt(40) = 0.174 Standard Deviation of The Population Essay Assignment

ME = 2.704 * 0.174 = 0.471

Therefore, the 99% confidence interval is:

CI = sample mean ± ME = 7.1 ± 0.471 = (6.629, 7.571)

Question 3

According to the findings of the hypothesis test’s “T-Stat” column, the value of t is 2.9268664. We compare the absolute t-statistic value (2.9268664) to the critical t-value (-1.7458837) in the “Critical t” column using an alpha of 0.05. We reject the null hypothesis H0 because the t-statistic is higher than the threshold t-value, and we come to the conclusion that there is enough evidence to support the alternative hypothesis HA, which states that the genuine population mean resting HR is less than 70. The sample mean, standard error, and t-value can be used to derive the 95% confidence interval for the population’s average resting heart rate. We can be 95% confident that the genuine population mean resting heart rate lies within the range of 70.1555 and 80.7857 because the confidence interval is (70.1555, 80.7857).This indicates that the study’s patient population may need further medical care because their resting heart rate was much higher than the norm for the general population. Standard Deviation of The Population Essay Assignment

Question 4

With a t-value of 2.93 and a p-value of 0.005, the one sample t-test results show strong evidence that the genuine population mean resting HR for the patient group in this study is fewer than 70 beats per minute. The alternative hypothesis HA:μ< 70 is accepted instead of the null hypothesis H0: μ= 70.

Question 5

Because the dependent variable is dichotomous (yes/no) and the independent variable only has two groups (normal vs. high), the independent groups t-test is incorrect in this case. For assessing this kind of data, a logistic regression or chi-square test might be preferable.

1. The rebellious organizations because the dependent variable is continuous and the independent variable has two distinct groups (low dose vs high dose), the t-test is acceptable in this case (white blood cell count).
2. Because there are three groups in the independent variable, the independent groups t-test is incorrect in this circumstance (low vs normal vs high). This kind of data analysis would be better served by the Kruskal-Wallis test or one-way ANOVA.
3. The rebellious organizations As the dependent variable is continuous and the independent variable contains two distinct groups (single vs. married), the t-test is acceptable in this case (happiness measured on a scale from 1 to 50). Standard Deviation of The Population Essay Assignment

Question 6

a.The same group of comatose patients were assessed with and without massage, and the dependent variable was the patients’ intracranial pressure, hence the dependent groups t-test was appropriate in this case.

1. Because the independent variable in this case has two separate groups (home vs. hospital) and the dependent variable is only evaluated once, the dependent groups t-test is unsuitable (48 hours after surgery). This kind of data analysis would be better served by an independent groups t-test.
2. The dependent groups t-test is acceptable in this case since the dependent variable is body weight and the independent variable is the same prisoner measured at two separate time points (1 month and 6 months following incarceration).
3. Because the dependent variable is the amount of time spent sleeping at night and the independent variable is the same group of narcoleptic patients assessed before and after nap therapy, the dependent groups t-test is applicable in this case. Standard Deviation of The Population Essay Assignment

Question 7

We can use the following formula to determine the t-statistic for independent groups:

t = (x1 – x2) / sqrt(s1^2/n1 + s2^2/n2) where s1 and s2 are their standard deviations, n1 and n2 are their sample sizes, and x1 and x2 are the means of Groups 1 and 2.

By entering the specified values, we obtain:

t = (78.1 – 75.1) / sqrt((6.1^2/25) + (5.7^2/25))

= 3 / sqrt(0.91984 + 0.80604)

= 3 / sqrt(1.72588)

= 3 / 1.3135

= 2.2835

The following formula can be used to compute the degrees of freedom for the t-test:

df = (s1^2/n1 + s2^2/n2)^2 / [(s1^2/n1)^2/(n1-1) + (s2^2/n2)^2/(n2-1)]

By entering the given values, we obtain:

df = (6.1^2/25 + 5.7^2/25)^2 / [(6.1^2/25)^2/24 + (5.7^2/25)^2/24]

= 1.72588^2 / [0.95644 + 0.83271]

= 1.72588^2 / 1.78915

= 1.6598

The crucial t-value for a two-tailed test with 48 degrees of freedom and =0.05 is roughly 2.011, according to a t-distribution table. We may reject the null hypothesis and come to the conclusion that there is a significant difference between the mean pain ratings of Groups 1 and 2 at =0.05 because our computed t-value of 2.2835 is greater than 2.011. Standard Deviation of The Population Essay Assignment

Question 8

According to the results of the independent groups t-test, group 1 cancer patients reported much more pain than group 2 cancer patients who underwent specialized massage treatments. The crucial t-value of 2.011 was exceeded by the computed t-value of 2.2835 with 48 degrees of freedom and 0.05, providing support for rejecting the null hypothesis.

Question 9

We need to compare a t-value to a significant t-value from a t-distribution table, based on the degrees of freedom (df) and alpha (Î) level, in order to establish whether it is statistically significant.

2. The significant t-value for t = 2.90, df = 25, and Î = 0.01 is 2.787. The t-value is statistically significant for a two-tailed test at the 0.01 alpha level since |2.90| > 2.787.

 

2. The critical t-value for t = 2.09, df = 25, and Î = 0.05 is 2.064. The t-value for a two-tailed test is not statistically significant at the 0.05 alpha level because |2.09| 2.064.
3. The significant t-value for t = 2.51, df = 10, and Î = 0.01 is 3.169. The t-value for a two-tailed test is not statistically significant at the 0.01 alpha level because |2.51| 3.169.
4. The significant t-value for t = 2.02, df = 150, and Î = 0.05 is 1.977. The t-value is statistically significant for a two-tailed test at the 0.05 alpha level since |2.02| > 1.977. Standard Deviation of The Population Essay Assignment

Question 10

ANOVA is a good choice in this circumstance. As there is just one independent variable (ethnicity) with several levels and one continuous dependent variable, a one-way ANOVA would be employed specifically (serum cholesterol levels).

1. In this case, ANOVA is appropriate. As there is just one independent variable (smoking status) with many levels and one continuous dependent variable, a one-way ANOVA would be employed specifically (health-related hardiness).
2. In this case, ANOVA is appropriate. Since there are two independent factors (maternal breast feeding status and ethnicity) and one continuous dependent variable, a two-way ANOVA would be utilized specifically (maternal bonding with infant).
3. In this case, ANOVA is appropriate. When the same individuals are utilized for each treatment and several measurements of the dependent variable (myocardial oxygen consumption) are made before and after each therapy, a repeated-measures ANOVA would be applied specifically. Standard Deviation of The Population Essay Assignment

 

 

 

Question 11

The F-statistic is 2.851 and the p-value is 0.097, according to the ANOVA table. We are unable to reject the null hypothesis that the population means are equal since the p-value is higher than the alpha level of 0.05. In other words, there is insufficient data to support the claim that scores on the PSS differ significantly among non-smokers, smokers, and quitters.

Question 12

The following theories will be tested:

H0: The three nations (Guinea, Liberia, and Sierra Leone) have comparable mean ages.

H1: At least one country’s mean age differs from the others.

F in the ANOVA table has a value of 2.2688964.

The crucial F-value with 2 and 491 degrees of freedom is 3.00 with an alpha of 0.05. (from an F-distribution table). We are unable to reject the null hypothesis since the computed F-value (2.2688964) is below the necessary F-value (3.00). This indicates that, at the 5% significance level, we lack adequate data to draw the conclusion that there is a significant difference in mean ages between the three countries. Standard Deviation of The Population Essay Assignment

Question 13

a.F=2.40, dfb=4, dfw=40, and =0.05

We must compare the F value to the significant value of F for the specified degrees of freedom in order to establish whether F is statistically significant at = 0.05. (dfb and dfw).

The significant F value for = 0.05, dfb = 4, and dfw = 40 is 2.48 when using a F table or calculator.

We fail to reject the null hypothesis because the computed F value (2.40) is less than the critical F value (2.48), which leads us to believe that there is no discernible difference between the groups

5. F=5.02, dfb=3, dfw=60, and =0.001

For = 0.001, dfb = 3, and dfw = 60, the critical F value is 4.05. We reject the null hypothesis and come to the conclusion that there is a significant difference between the groups since the calculated F value (5.02) is higher than the significant F value (4.05).

3. F=3.45, dfb=3, dfw=27, and =0.05.

3.10 is the significant F value for the conditions = 0.05, dfb = 3, and dfw = 27. We reject the null hypothesis and come to the conclusion that there is a significant difference between the groups since the calculated F value (3.45) is higher than the critical F value (3.10).

4. F = 4.99, DFW = 150, DFW = 2, and = 0.01

3.01 is the significant F value for the conditions = 0.01, dfb = 2, and dfw = 150. We reject the null hypothesis and come to the conclusion that there is a significant difference between the groups since the calculated F value (4.99) is higher than the critical F value (3.01).

2 pts) Population A and Population B both have a mean height of 71.0 inches with an SD of 3.2. A random sample of 30 people is picked from population A, and random sample of 55 people is selected from Population B. Which sample mean will probably yield a more accurate estimate of its population mean? Standard Deviation of The Population Essay Assignment

2. (2 pts) Suppose we obtained data on vein size after application of a nitroglycerin ointment in a sample of 40 patients. The mean vein size is found to be 7.1 mm with an SD of 1.1. Using a t distribution table, what are the confidence limits for a 95% confidence interval? For a 99% confidence interval?

3. (2 pts) In a pilot study evaluating the use of a new drug to lower resting heart rates (HR) of patients, the following data was recorded:
Subject # Resting HR
001 72
002 88
003 71
004 87
005 64
006 79
007 79
008 69
009 66
010 68
011 78
012 89
013 71
014 81
015 77
016 75
017 69

Given that the average resting HR of the general population for this study is 70, use StatCrunch to perform the appropriate t test. What is the value of t? Using an alpha of 0.05, is the t statistic significant? Why? What are the confidence limits for a 95% confidence interval here and what do they mean for this patient group? Copy and Paste your work from StatCrunch into your Word document submission. Standard Deviation of The Population Essay Assignment

4. (1 pt) Write one or two sentences that could be used to report the results obtained for the t-test in Exercise 3 above (be sure to state your hypotheses).

5. (1 pt) For which of the following situations is the independent groups t-test appropriate (if inappropriate, why?):
a. The independent variable is infant birth weight at one week (normal vs high); the dependent variable is survival to 1 year (yes/no).
b. The independent variable is radiation treatment on the same throat cancer patients (after a low dose and then a high dose treatment); the dependent variable is white blood cell count.
c. The independent variable is infant birth weight (low vs normal vs high); the dependent variable is average weight in first grade.
d. The independent variable is married status (single vs married); the dependent variable is happiness measured on a scale from 1 to 50.

6. (1 pt) For which of the following situations is the dependent groups t-test appropriate (if not appropriate, why?)
a. The independent variable is massage therapy to comatose patients (same patients measured with and then without massage); the dependent variable is the patients†intracranial pressure.
b. The independent variable is birth type (home vs hospital); the dependent variable is blood pressure of the infant 48 hours after surgery.
c. The independent variable is time since incarceration on the same inmate (repeat offender), 1 month vs 6 months; the dependent variable is body weight.
d. The independent variable is nap therapy for narcoleptics (same patients before vs after treatment); the dependent variable is the length of time for sleep at night.

7. (2 pts) Suppose we wanted to test the hypothesis that a control group of cancer patients (Group 1) would report higher mean pain ratings than an experimental group receiving special massage treatments (Group 2). Use the following information (mean, standard deviation, and sample size). Compute a t-statistic for independent groups:

mean group 1 = 78.1 SD1 = 6.1 n1 = 25
mean group 2 = 75.1 SD2 = 5.7 n2 = 25

What are the degrees of freedom and the value of t ? Using α=0.05 for a two-tailed test, is this t statistic significant? Show your calculations (or StatCrunch output) for full credit.

8. (1 pt) Write one or two sentences that could be used to report the results obtained for the t-test in Exercise 7 above (be sure to state your hypotheses).

9. (1 pt) For each of the following t values, indicate whether the t is statistically significant for a two-tailed test, at the specified alphas: Standard Deviation of The Population Essay Assignment

a. t = 2.90, df = 25, α = 0.01
b. t = 2.09, df = 25, α = 0.05
c. t = 2.51, df = 10, α = 0.01
d. t = 2.02, df = 150, α = 0.05

10. (2 pts) For each of the following situations that will include one or more independent variables (IV) and a dependent variable (DV), indicate whether ANOVA is appropriate. If not appropriate, the reason why not; and, if appropriate, the type of ANOVA that would be used (i.e., one-way, repeated measures, etc.).
a. The IV is ethnicity (Asian, White, African American, Hispanic); the DV is serum cholesterol levels.
b. The IV is smoking status – smokers vs non-smokers vs quitters; the DV is health-related hardiness as measured on a 20-item scale.
c. The IV is maternal breast feeding status (breastfeeds daily vs breastfeeds at least 1-3 times/week vs doesnâ€t breastfeed) and ethnicity White vs Non-White); the DV is maternal bonding with infant, as measured on a 20-item self-report scale.
d. The IV is treatment group for patients with drug-induced shivering (extremity wraps vs high room temp vs normal room temp without wraps) same patient used for each treatment; the DV is myocardial oxygen consumption, measured before and after each treatment. Standard Deviation of The Population Essay Assignment

11. (2 pts) Suppose we wanted to compare the somatic complaints (as measured on a scale known as the Physical Symptom Scale or PSS) of three groups of people: non-smokers, smokers, and people who recently quit smoking. Using the following data for PSS, do a one-way ANOVA to test the hypothesis that the population means are equal:

Nonsmokers Smokers Quitters
19 31 30
28 29 21
17 22 27
20 29 29
25 28 22

Using StatCrunch, determine the means for the three groups and compute the sums of squares, degrees of freedom, and mean squares for these data. What is the value of F? Using an alpha of 0.05, is the F statistic significant? Why? Copy and Paste your work from StatCrunch into your Word document submission.

12. (2 pts) Look back at the data set in Excel we analyzed in Unit 1 for discussion question 2. This was the simple random sample (n=494) taken from ebola cases that you will also find linked to this assignment in the course shell assignment area. Using StatCrunch, perform an ANOVA on the ages of the cases with respect to country of origin. Determine the mean age for the three groups (countries) and compute the sums of squares, degrees of freedom, and mean squares for these data. What are the hypotheses to be tested here? What is the value of F? Using an alpha of 0.05, is the F statistic significant? Why or why not? Copy and Paste your work from StatCrunch into your Word document submission for full credit. Standard Deviation of The Population Essay Assignment

13. (1 pts) For each of the following F values, indicate whether the F is statistically significant at the specified alpha level:

a. F = 2.40, dfb = 4, dfw = 40 α = 0.05
b. F = 5.02, dfb = 3, dfw = 60, α = 0.001
c. F = 3.45, dfb = 3, dfw = 27, α = 0.05
d. F = 4.99, dfb = 2, dfw = 150, α = 0.01 Standard Deviation of The Population Essay Assignment