Biostats Final Answers

28. Parameters are characteristics of a population, are fixed for that population though inevitably unknown in statistical problems.TRUE 29. The standard error of the mean increases with decreasing sample size.TRUE 30. The standard error of the mean increases with increasing population size.TRUE 31. The Standard Deviation for a population of measurements 16.0 meters, the standard deviation computed from a sample of sixteen observations from this population would be expected to be approximately 4.0 meters.FALSE 32. The Standard Deviation for a population of measurements 25 mg, the standard deviation computed from a sample of 625 observations from this population would be expected to be approximately 1.0 mg. FALSE 33. If in a sample of size 2 the observations are 0 and 2, the variance is 2 . √ FALSE 34. One can obtain a true random sample of the individuals in a community by using the information from the Department of Social Security. FALSE 35. One can obtain a true random sample of the individuals in a community by using the information from the Department of Motor Vehicles. FALSE 36. One can obtain a true random sample of the individuals in a community by using the voter registration database. FALSE 37. Individual arithmetic means computed from random sample can vary among themselves but are usually equal to the mean of the population. FALSE 38. Arithmetic means computer from random samples vary among themselves and may differ from the mean of the population. TRUE 39. In any skewed distribution, mean > median. FALSE 40. After subtracting the median from each observation in a sample, the sum of the values is always zero. FALSE Old Midterm Questions In testing H0, μ=100 with α = 0.05 and n=50 a) If n is change to 15 and α is left unchanged, what happens to β? INCREASED b) If n is change to 100 and α is left unchanged, what happens to 1-β? INCREASED c) If α is changed to 0.01 and n is left unchanged, what happens to β? INCREASED d) If H0 is not correct, but the true μ is actually very close to 100, β should be close True/False Question Bank 1. The smaller the level of confidence, the shorter the confidence interval. TRUE 2. For a fixed confidence level, when the sample size increases, the width of the confidence interval for the population mean, μ, decreases. TRUE 3. For a fixed confidence level, when the sample size increases, the margin of error of the confidence interval for a population mean decreases. TRUE 4. The null hypothesis H0 is considered correct until proven otherwise. TRUE 5. If the null is rejected, this means that the null is not true. FALSE 6. If the null is not rejected, this means that the null hypothesis is true. FALSE 7. As the sample size increases, the power of the test of the null hypothesis, H 0, increases. TRUE 8. α = 1 – β FALSE 9. If we reject H0, then we can say we have conclusively proven H1 is true. FALSE 10. If we reject H0, then we can say we have conclusively proven H0 is false. FALSE 11. If we take a large enough sample from any population, then the histogram for the sample data would be expected to be bell-shaped. FALSE 12. If we take a large random sample, the mean of the sample will be approximately equal to the median. FALSE 13. If we reject H0 at α = 0.05, the probability is 0.95 that H1 is correct. FALSE 14. The larger the sample size, the larger the confidence interval. FALSE 15. The alternative hypothesis, H1, is considered correct until demonstrated otherwise. FALSE 16. As the sample size decreases, the probability of a β error decreases. FALSE 17. In systematic random sampling, any selected sequence of observation is just as likely to be chosen as any other distinct sequence. TRUE 18. In stratified random sampling, we define strata so that the individuals in each of these strata are as heterogeneous as possible. FALSE 19. In any symmetric distribution the mean, median, and mode have the same value. FALSE 20. Since x-bar is an unbiased estimate of population mean, this means that any value of c-bar that we observe must be very close to the population mean. FALSE 21. The sample standard deviation is used with skewed data as a measure of variability of the sample.FALSE 22. The sample midrange can be used as a reasonable estimate of the sample standard deviation.FALSE 23. Subtracting the arithmetic mean from each observation in a sample, the sum of values is always zero. TRUE 24. Subtracting the arithmetic mean from each observation in a sample, results in a group of values whose mean is always zero.TRUE 25. Subtracting the arithmetic mean from each observation in a sample, results in a group of values whose standard deviation is always zero.FALSE 26. If X1=3, X2=81, X3=27, X4=3, and X5=729, then the geometric mean is 27. TRUE 27. Parameters are characteristics of a population, are fixed for that population, but are known under most circumstances. to: 0, 1, or 0.5? 1 e) If H0 is not correct, but the true μ is actually very far from 100, β should be very close to: 0, 1, or 0.5? 0 If our significance level, α, is set equal to 1, then we always reject H0. If we sample the entire population, then α=β=0. If we set α=0.05, then β must be… Can’t tell with the information given. Suppose we test a new device in 20 healthy volunteers and measure the change in their heart rate. The mean was 10 and the standard deviation was 5. Compute the 90% confidence interval for μ , the population mean change in the heart rate. x±t s → 10 ±1.729 5 →( 8.067 , 11.933 ) ; 8.067< μ<11.93 √ 20 √n t-value: df=19, Critical Value = 0.05 (90%) Which of the following are true about the interval you calculated and the calculations used? 90% Confidence Interval, x=12 , s=5 , n=16 (i) P(calculated interval is correct) = 0.90 (ii) P(calculated interval is not correct) = 0.10 (iii) The t-values used to calculate the interval were ± 1.753 (iv) The z-values used to calculate the interval were ± 1.64 (v) If we took another 16 volunteers and calculated another interval, we would expect to get the exact same numerical values. (vi) The formula we used to calculate our interval gives us the right answer (contains μ ) 90% of the time. (vii) If we had used a 95% confidence level, the width of the interval would have been smaller. An industrious Biostatistics 100A student decides to compute a condifence interval for the mean daily caloric intake of UCLA students. As such, she takes a sample of 100 students and obtains x-bar = 1900 and s=100. The interval was then calculated and the result was: 1642 <μ < 2158. What level of confidence was used? ME= 1000 2158−1642 1000 → ME=t =25 =258:1900 ±t 2 100 √ √ 100 t = 2.58 with df = 99  Critical Value = 0.005; CI = 0.005 x 2 = 0.001  99% If we had wanted the entire width of this interval to be 129, what sample size should we have used to achieve the same level of confidence that was used in part a? ME= s 129 1000 → n=160 =64.5→ ME=t → 64.5=2.58 2 √n √n Assume for the purposes of just this portion of the problem that the confidence level of the interval given in part a is 90%, then which of the following statements are correct? (i) P ( μ <1642 )+ P ( μ>2158 ) =0.10 (ii) P ( 1642< μ < 2158) =0.90 (iii) If we were to take another sample from this population, we would probably get an interval with different endpoints. (iv) The probability is 90% that if we were to take another sample, the new x-bar would be between 1642 and 2158. (v) The formula that generated the interval in question yields a correct interval (contains μ ) 90% of the time. Assume that the length of time between charges of a cellular phone is normally distributed with a mean of 10 hours and a standard deviation of 5 hours. If I have a fully charged phone at 8 am, find the probability that I will have to recharge the phone by 10 pm (same day). By 10 pm : x=14 hours→ Z= x−μ 14 −10 =+ 0.80→ = 5 σ p-value= 0.7881 or 78.81% If I take a sample of 16 phones, what is the probability that the average length of time between charges for these phones is less than 8 hours? x=8 hours → Z= x−μ 8−10 =−1.60 → = 5 σ p-value= 0.0548 or √n √16 Since p-value > α , we fail to reject H 0 , there is not significant evidence to support the claim that (Write out H a substituting words for μ ) Risk of Error: If reject H0, risk of Type I Error (α Error) If do not reject H0, risk of Type II Error (β Error) Calculations for 1-Sample Z/T Confidence Interval x ± Z¿ x ± ME σ SE= √n 5.48 % σ x±t¿ or √n s √n x ± Z P SE ME =Z P SE df =n−1 α Critical Value (p*) = 1 α=12 confidence level 100 σ σ x−Z ¿ < μ< x +Z ¿ or √n √n s σ x−t ¿ < μ μ2 ¿ (One-Tailed) OR H a : The mean of population 1 is not the same as the mean of population 2 ( μ1 ≠ μ2 ¿(Two-Tail) Assumptions: 1. The sample was randomly selected, if not stated 2. If n > 30, use CLT (Applies to both samples n1 and n2 ) 3. If n < 30, assume normal distribution (Applies to both samples n1 and n2 ) 4. σ is known (for Z Test) 5. Two samples are independent The Decision Rule: Reject Ho if the p-value is less than α = 0.05 or, for this case, if the student t-value Z/T > value and Z/T < - value 1-Sample Z/T Hypothesis Test Hypothesis Examples: Calculations: H 0 : The strict vegetarian diet will have the same average cholesterol levels compared to the general population aged 30-49, 180 mg/dl. μ=180mg / dl H a : The strict vegetarian diet will have lower average cholesterol levels compared to the general population aged 30-49, 180 mg/dl. μ<180 mg / dl (One-Tailed) Z= OR √ x 1−x 2 x1 −x2 t= σ1 2 σ 22 s1 2 s 22 + + n1 n 2 n1 n2 df =n1 + n2−2 √ H a : The strict vegetarian diet will have different average cholesterol levels compared to the general population aged 30-49,2 180 mg/dl. 2 σ1 σ2 SE ¿ μ ≠180 mg / dl (Two-Tailed) + n 1 n2 Assumptions: √ 1. The sample was randomly selected, if not stated 2. If n > 30, use CLT 3. If n < 30, assume normal distribution 4. σ is known (for Z Test) P-Value (One Tail) x −μ σ √n t= P-Value (One Tail) x−μ s √n df =n−1 P-Value X 2 (Two Tail) α, we reject H 0 , there is significant evidence to support the μ) claim that (Write out H a substituting words for 2 2 + SE 2 P-Value X 2 (Two Tail) Since p-value ≤ α , we reject H 0 , there is significant evidence to support the Interpretation: Since p-value ≤ = SE1 Interpretation: The Decision Rule: Reject Ho if the p-value is less than α = 0.05 or, for this case, if the student t-value Z/T > value and Z/T < - value Calculations: Z= 2 SEOverall claim that (Write out H a substituting words for μ) Since p-value > α , we fail to reject H 0 , there is not significant evidence to support the claim that (Write out H substituting words for μ ) a Risk of Error: If reject H0, risk of Type I Error (α Error) If do not reject H0, risk of Type II Error (β Error) Calculations for 2-Sample Z/T Confidence Interval 2 2 σ1 σ 2 ¿ or x 1−x 2 ± Z + n 1 n2 √ √ 2 1 2 1 1 Assumptions: 1. The sample was randomly selected, if not stated 2. Expected Frequency is greater than or equal to 5 The Decision Rule: 2 s1 s2 + n1 n2 ¿ ¿ – Z SE ( y – y ) <( μ 1 – μ 2 ) <+Z SE ( y – y ) – t ¿ SE ( y – y )< ( μ 1 – μ 2) <+t ¿ SE ( y – y x 1−x 2 ± Z ¿ ± t ¿ 2 or 2 1 Reject Ho if the p-value is less than α = 0.05 or, for this case, if (Insert 2 ) χ 2 χ 2is greater than Value) Calculations: k χ 2= ∑ ¿ ¿ ¿ Interpretation for 2-Sample Z/T Confidence Interval “We are % confident that the difference between the mean of the (insert population 1) and the mean of the (insert population 2) is between (lower limit value) and (upper limit value).” (Look at Paired T-Test for descriptive) IF YOU WANT TO BE SIGNIFCANT, CALUCLATE BOTH TEST AND CI!! P-Value (Two Tail) i=1 P-Value ÷ 2 (One Tail) Interpretation: Since p-value ≤ α , we reject H 0 , there is significant evidence to support the Paired T Hypothesis Test claim that (Write out H a substituting words for Hypothesis Examples: Since p-value > α , we fail to reject H 0 , there is not significant evidence to H 0 : The mean of (Insert population 1) is the same as the mean of (Insert Population 2) μ D=0 H a : The mean of (Insert population 1) is the same as the mean of (Insert Population 2) μ D ≠0 (Two-Tailed) Assumptions: 1. The sample was randomly selected, if not stated 2. If n > 30, use CLT 3. If n < 30, assume normal distribution 4. σ is known (for Z Test) 5. Two samples are independent The Decision Rule: Reject Ho if the p-value is less than α = 0.05 or, for this case, if the student t-value Z/T > value and Z/T < - value Calculations: D = Y 1−Y 2 t s= d−0 SED t s= OR √n D df =n D−1 P-Value (One Tail) d−0 sD P-Value X 2 (Two Tail) Interpretation: H 0 , there is significant evidence to support the claim that (Write out H substituting words for μ ) a Since p-value > α , we fail to reject H 0 , there is not significant evidence to support the claim that (Write out H a substituting words for μ ) Since p-value ≤ α, we reject Risk of Error: If reject H0, risk of Type I Error (α Error) If do not reject H0, risk of Type II Error (β Error) Calculations for Paired t-Test Confidence Interval d ±t d −t ¿ ¿ sD √ nD sD √ df =n D−1 < μ D < d+t ¿ sD √ nD nD Interpretation f or Paired t-Test Confidence Interval We are % confident that the mean difference between (insert population 1) and (insert population 2) is between (lower limit value) and (upper limit value). The null value of the confidence interval for the mean difference is zero. Therefore, based on the % confidence interval, we can conclude that there is no statistically significant difference in (Insert Measure), because the confidence interval for the mean difference includes zero. OR We are % confident that the mean difference between (insert population 1) and (insert population 2) is between (lower limit value) and (upper limit value). The null value of the confidence interval for the mean difference is zero. Therefore, based on the % confidence interval, we can conclude that there is statistically significant difference in (Insert Measure), because the confidence interval for the mean difference does not include zero Chi Squared Hypothesis Test Hypothesis Examples: H 0 : The population proportion of (Insert category 1), (Insert category 2), etc. are equal. P1 =. P2=. Pn H a : At least onePi is not equal to its hypothesized value. μ) support the claim that (Write out H a substituting words for μ ) Risk of Error: If reject H0, risk of Type I Error (α Error) If do not reject H0, risk of Type II Error (β Error)