Point Estimates and Critical Values
An interval for a proportion is (0.02, 0.13). What is the point estimate and margin of error.
P-hat= 0.075
ME=0.055
A new Covid vaccine is being tested. A random sample of 350 US adults are given the vaccine and observed over the next year to see if they develop Covid.55 out the 350 did get Covid that year. If an interval was constructed to estimate the true proportion of those who would get Covid with vaccine, what is the standard error? Do not round the point estimate.
SEp-hat=0.0195
A French study was conducted in the 1990s to compare the effectiveness of using an instrument called a cardiopump with the effectiveness of using traditional cardiopulmonary resuscitation (CPR) in saving lives of heart attack victims. Heart attack patients in participating cities were treated with either a cardiopump or CPR, depending on whether the individual’s heart attack occurred on an even-numbered or an odd-numbered day of the month. Before the start of the study, a coin was tossed to determine which treatment, a cardiopump or CPR, was given on the even-numbered days. The other treatment was given on the odd-numbered days. In total, 754 patients were treated with a cardiopump, and 37 survived at least one year; while 746 patients were treated with CPR, and 15 survived at least one year. Perform a statistical test to determine whether the survival rate for patients treated with a cardiopump is significantly higher than the survival rate for patients treated with CPR. What is the null and alternative hypothesis? Define parameters.
P1: proportion of those treated with a cardio pump
P2: Proportion of those treated with CPR
H0: P1-P2=0
Ha: P1-P2>0
A random sample was collected. 32 out of 75 people responded yes to the question. If the H0: P=0.49 and Ha: P≠0.49 what is the test statistic?
z=-1.10
A research study gives a 95% confidence interval for the proportion of dogs helped by a new anti-inflammatory drug was (0.56, 0.65). A random selection of 150 dogs with arthritis were studied. Interpret the interval in context.
We are 95% confident that the interval 56% to 65% captures the true proportion of dogs helped by this new anti-inflammatory drug.
UW Wisconsin conducted a survey in 2020 and in 2024. A random sample of 400 residents of Wisconsin each of the years. 27% of residents in 2020 said the country was headed in the right direction. 33% said the same in 2024. What is the point estimate for the percent increase from 2020 to 2024 (P1=2024, P2=2020)?
(p-hat1-p-hat2)=0.06
A new Covid vaccine is being tested. A random sample of 350 US adults are given the vaccine and observed over the next year to see if they develop Covid.55 out the 350 did get Covid that year. If an 90% confidence interval was constructed to estimate the true proportion of those who would get Covid with vaccine, what is the margin of error? Do not round the point estimate.
ME= 0.0320
Some boxes of a certain brand of breakfast cereal include a voucher for a free video rental inside the box. The company that makes the cereal claims that a voucher can be found in 20 percent of the boxes. However, based on their experiences eating this cereal at home, a group of students believes that the proportion of boxes with vouchers is less than 0.2. This group of students purchased 65 boxes of the cereal to investigate the company’s claim. The students found a total of 11 vouchers for free video rentals in the 65 boxes.
Suppose it is reasonable to assume that the 65 boxes purchased by the students are a random sample of all boxes of this cereal. Based on this sample, is there support for the students’ belief that the proportion of boxes with vouchers is less than 0.2 ? Provide statistical evidence to support your answer. Set up null and alternative hypotheses. Define parameter.
P:proportion of cereal boxes that contain vouchers
H0: P=0.20
Ha: P<0.20
A random sample was collected. 32 out of 75 people responded yes to the question. If the H0: P=0.49 and Ha: P≠0.49 what is the p-value?
p-value=0.2726
A research study gives a 95% confidence interval for the proportion of dogs helped by a new anti-inflammatory drug was (0.56, 0.65). A random selection of 150 dogs with arthritis were studied. Interpret the confidence level in context.
If you take many samples of size 150 dogs from this population, approximately 95% of the intervals will capture the true proportion of dogs helped by this new anti-inflammatory drug.
A 98% confidence is constructed. A sample size of 100 will be used. Assuming the population which the sample is being drawn from is approximately normal, what is the upper critical value needed to construct the interval?
z98=2.32
What are two ways to decrease the margin of error?
Increase sample size and decrease the confidence level.
University of Florida is worried it may not have enough housing for incoming freshman. It is expensive to build new housing so they will operate under the assumption that there is enough housing. A study is conducted. What would be a Type 1 error if the Null Hypothesis is that there is enough housing and the Alternative Hypothesis is that there is not enough housing? What would be the repercussions of making this type 1 error?
The conclusion would be that there is not enough housing when there really was. They would spend money on a problem that doesn't exist.
2 independent random samples were collected. Population one had 48 out of 150 responded favorably to a question and 76 out of 170 from population 2 responded favorable to that same question. What is the test statistics if the H0: P1-P2=0 and H0:P1-P2<0?
z=-2.33
A random sample was collected. 32 out of 75 people responded yes to the question. The H0: P=0.49 and Ha: P≠0.49. The test yielded a p-value of 0.2726. At a significance level of 0.10, what would be the conclusion to your study?
Since the p-value (0.2726) is greater than the alpha(0.10) we fail to reject the null hypothesis. We do not have evidence the true proportion is different than 49%.
A 89% confidence interval is constructed. A sample size of 100 will be used. Assuming the population which the sample is being drawn from is approximately normal, what is the upper critical value needed to construct the interval?
z89=1.60
You are going to construct a 95% confidence interval for a population proportion and want a margin of error no more than 0.05. Historical data indicates the population proportion has remained constant at about 0.65. What is the minimum sample size needed to construct the interval.
n≥350
University of Florida is worried it may not have enough housing for incoming freshman. It is expensive to build new housing so they will operate under the assumption that there is enough housing. A study is conducted. What would be a Type 2 error if the Null Hypothesis is that there is enough housing and the Alternative Hypothesis is that there is not enough housing? What would be the repercussions of making this type 2 error?
They would conclude that there is enough housing when there is not. They would not have enough housing for incoming freshman.
2 independent random samples were collected. Population one had 48 out of 150 responded favorably to a question and 76 out of 170 from population 2 responded favorable to that same question. What is the p-value if the H0: P1-P2=0 and H0:P1-P2<0?
p-value=0.0010
2 independent random samples were collected. Population one had 48 out of 150 responded favorably to a question and 76 out of 170 from population 2 responded favorable to that same question. This study produced a p-value of 0.0010 and H0: P1-P2=0 and H0:P1-P2<0. At significance level of 0.05 what would your conclusion of the study be?
Since our p-value (0.0010) is less than the alpha (0.05), we reject the null hypothesis. Therefore we do have evidence that the P2 is less than P1.
The interval for a population proportion (0.119, 0.195) was constructed. A sample size of 350 was taken. What critical value and confidence level was used to construct this interval? Must provide work to support answer.
Z=1.96
CL=95%
A researcher wants to investigate the difference in proportion of marriages that end in divorce in Florida and in Wisconsin. How large of a sample (same for each state) should be taken to estimate the difference within 6% at 92% confidence? Show your work.
n1 and n2 must be at least 426.
What is the probability of making a Type 1 error if the significance level of 0.05?
0.05
If we had a p-value for a left-tailed test of 0.012, what would the p-value be for the same data but now a two-tailed test?
p-value=0.024
If the p-value from a two-tailed test was 0.0345, what would the p-value be if it was changed to a right-tailed test and all other information remained the same?
p-value=0.01725