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Suppose that 47% of all adult women think they do not get enough time for themselves. An opinion poll interviewed 1025 women and records the sample proportion. Describe the shape and find the mean and standard deviation of the sampling distribution.
Shape: Normal Distribution: The sampling distribution of the sample proportion will approximate a normal distribution due to the large sample size (n = 1025) and the Large Count Conditions will be met.
Mean: 0.47
Standard Deviation: approx 0.0155. .
The mean monthly fee that households pay service providers for Internet access is $28 with a standard deviation of $10, but the distribution is not normal. A sample survey asks a SRS of 500 households with Internet access how much they pay. What is the probability that the average fee paid by the sample households exceeds $29?
0.0125 OR 1.25%
A mail order company advertises that it ships 90% of its orders within 3 working days. You select a SRS size 100 of the 500 orders received in the past week for an audit. The audit shows 86 of these orders were shipped on time. What is the probability that 86 or fewer in a SRS of 100 are shipped on time?
0.0918 OR 9.18%
The average age of cars owned by residents of a small city is 4 years with standard deviation of 2.2 years. An SRS of 400 cars is selected, and the sample mean age computed. Describe the shape, the mean and standard deviation of the sampling distribution.
Shape: Normal Distribution: The sampling distribution of the sample mean will approximate a normal distribution due to the large sample size (n = 400) and the Central Limit Theorem.
Mean: 4 years.
Standard Deviation: approx 0.11 years.
The incomes in a certain large population of college teachers have a normal distribution with mean $35,000 and standard deviation $5000. Four teachers are selected at random from this population to serve on a salary review committee. What is the probability that their average salary exceeds $40,000?
0.0228 OR 2.228%
The strength of paper coming from a manufacturing plant is known to be 25 pounds/square inch with a standard deviation of 2.3. In a random sample of 40 pieces of paper, what is the probability that the mean strength is above 25.5 pounds/square inch?
0.0853 OR 8.53%
A promoter knows that 23% of males enjoy watching boxing matches. In a random sample of 125 men, what is the shape, mean and standard deviation of the sampling distribution?
Shape: Normal Distribution due to passing the Large Count Conditions.
Mean: 0.23 (23%).
Standard Deviation: approx 0.0376.
A promoter knows that only 12% of females enjoy watching boxing matches. In a random sample of 125 women, what is the probability that more than 10% of the females enjoy watching boxing matches?
0.7549 OR 75.49%
The average number of missed school days for students in public schools is 8.5 with a standard deviation of 4.1. In a sample of 200 public school students, what is the probability that the average number of days missed is less than 8 days?
0.0409 OR 4.09%
A SRS of 100 Americans found that 61% were satisfied with service provided by the dealer from which they bought their car. Describe the shape, mean, and standard deviation of the sampling distribution.
Shape: Normal Distribution due to Large Count Conditions.
Mean: 0.61 (61%).
Standard Deviation: approx 0.049.
A study of voting chose 663 registered voters at random shortly after an election. Of these, 72% said they had voted in the election. Election records show that only 56% of registered voters voted in the election. Identify the population, parameter, sample, and statistic of this study.
Population: All registered voters.
Parameter: Proportion of registered voters who voted (56%). Sample: 663 registered voters surveyed.
Statistic: Proportion of surveyed voters who said they voted (72%).
A polling organization wants to estimate the proportion of voters who favor a new law banning smoking in public buildings. The organization decides to increase the size of its random sample of voters from about 1500 people to about 4000 people right before an election. What is the effect of this increase to the estimate and why?
Increasing the sample size to 4000 will lead to a more accurate estimate of the proportion of voters favoring the law. This is because a larger sample size decreases variability and provides a better representation of the population.
A machine is designed to fill 16-oz bottles of shampoo. When the machine is working properly, the amount poured into the bottles follows a normal distribution with mean 16.05 ounces and standard deviation of 0.1 ounces. Assume that the machine is working properly, and 4 bottles are randomly selected. What is the probability the mean of the 4 bottles weight will be less than 15.9 ounces?
If this were to happen to our sample, could we determine if the machine was working correctly or not?
0.0013 or 0.13%
Yes, we can. With the proportion being less than 5%, it indicates that the machine is likely not working correctly.
The average number of missed school days for students in public schools is 8.5 with a standard deviation of 4.1. In a sample of 20 public school students, what is the probability that the average number of days missed is less than 8 days?
0.2912 OR 29.12%
A report claimed that 20% of respondents subscribe to the “5-second rule”. Assume this figure is the true for the population of US adults. Let the proportion of people who subscribe to the 5-second rule in a SRS of 40 from this population. What can we determine about the shape, mean, and standard deviation of the sampling distribution?
Shape: Normal Distribution. Due to the Large Count Conditions being met.
Mean = 0.20 (20%)
Standard Deviation = 0.0632
A report claimed that 15% of respondents subscribe to the “5-second rule”. Assume this figure is the true for the population of US adults. Let P-hat = the proportion of people who subscribe to the 5-second rule in a SRS of 60 from this population. If we get a p-hat = 0.2, does this provide convincing evidence that the proportion of people that subscribe the “5-second rule” is higher than initially thought?
No. The proportion of them having a higher than 0.2 or 20% is 0.1401. Which is 14.01%. This is higher than 5% which makes it not statistically significant evidence.
A report claimed that 20% of respondents subscribe to the “5-second rule”. Assume this figure is the true for the population of US adults. Let p-hat = the proportion of people who subscribe to the 5-second rule in a SRS of 80 from this population. If we get a p-hat = 0.25, does this provide convincing evidence that the proportion of people that subscribe the “5-second rule” is higher than initially thought?
No. The proportion of respondents having a higher than 0.25 or 25% is 0.1314, which is 13.14%. This is higher than 5%, indicating insufficient evidence.
What does it mean to be a biased estimator?
A biased estimator is a statistic that systematically overestimates or underestimates the parameter it is intended to estimate. This can lead to inaccurate conclusions about the population based on the sample data.
What does it meant to be an unbiased estimator?
An unbiased estimator is a statistical estimator whose expected value equals the true value of the parameter being estimated. This means that, on average, it accurately estimates the parameter without systematic error.
A report claimed that 15% of respondents subscribe to the “5-second rule”. Assume this figure is the true for the population of US adults. Let p-hat = 0.4 be the proportion of people who subscribe to the 5-second rule in a SRS of 60 from this population. Describe the population, parameter, sample, and statistic of our study.
The population consists of all US adults,
the parameter is the true proportion of respondents who subscribe to the 5-second rule (15%),
the sample is the SRS of 60 individuals from this population, and
the statistic is the sample proportion (p-hat = 0.4).
What probability must we be below to have “convincing evidence” that our assumption/ given information is incorrect/ suspect?
A probability of less than 0.05 is typically required to have "convincing evidence" against the null hypothesis, suggesting that the assumption or given information may be incorrect.
