When to use T, Z, and Proportions
We use a t-test or interval when we:
Are given sample standard deviation instead of population standard deviation.
A company takes a sample of 40 employees. If the company wants to construct a confidence interval with 95% confidence, what critical value do they need?
t* = 2.023
State the two formulas for sample size as well as if they are for proportions or z intervals.
p-hat x (1-p-hat) x (z*/ME)2
(z* x population standard deviation/ME)2
Amy usually sees approximately 60 poppy plants survive after the last frost. The temperature was colder this year so she thinks she may have less than that. State the null and alternate hypothesis.
H0: mu = 60, H1: mu < 60)
A recent poll of 1500 new home buyers found that 60% hired a moving company to help them move. If we were to construct a 95% confidence interval, would you use z-interval, t-interval, or proportion formulas?
Proportion formulas.
Construct a 99% confidence interval for a sample of 145 with a mean of 780. Assume a standard deviation of 48.
(769.73, 790.27)
How large a sample would we need to estimate p with a margin of error of 0.04 with 95% confidence?
601
A test is made of H0: mu = 40 verses H1: mu > 40. A sample of size 26 is drawn. The sample mean and standard deviation are 46 and 10, respectively.
The average number of days absent per student per year at West Valley School District is 17 days with a population standard deviation of 4 days. The counselor believes that the mean is greater than that for Freshmen. If he conducts a hypothesis test at the 0.05 significance level, will he use a t-test, z-test, or proportion test?
You construct a 95% confidence interval on the number of miles per gallon a certain model of car will get. You get the interval (20.95, 23.56). Give an appropriate statement of confidence.
We are 95% confident that the population mean for the number of miles per gallon this model of car will get is between 20.95 and 23.56 miles per gallon.
A psychologist has developed a new test for spatial perception and she wants to estimate the mean score achieved by male pilots. How many people must she test if she wants a margin of error of 2 points with 95% confidence? Assume σ= 21.2
432 people
A library estimates that the average number of readers who prefer paper books over eBooks on a scale of 100 is 82. A publisher thinks that number is too high. He conducts a hypothesis test.
Type I error.
Suppose it is reported that the mean price of a gallon of milk is $3.53. A random sample of 35 retail establishments this year provides a mean price of $3.59 with a standard deviation of $0.26. Should we do a z-test, t-test, or proportion test for this problem?
T-test
A new car is being released by company A. They are testing to see if their gas mileage claims are accurate, so they take a SRS of 15 new cars and measure the gas mileage. Assume the sample is normally distributed. From this sample, they have a mean of 37.7 mpg and a standard deviation of 4.22 mpg. We want to find a 95% confidence interval for the population mean gas mileage for this new car.
(35.363, 40.037)
A university wants to conduct a survey of graduates from the past year to determine which proportion obtained a job in a field related to their degree. A similar study was conducted several years ago and found that 73% had obtained a job related to their degree. How many graduates must be surveyed to have a margin of error of 4% with a 95% confidence level?
474 graduates
The Fair Isaac Corporation (FICO) credit score is used by banks and other lenders to determine whether someone is a good credit risk. Scores range from 300 to 850, with a score of 720 or more indicating that a person is a very good credit risk. An economist wants to determine whether the mean FICO score is lower than the cut off of 720. She finds that a random sample of 100 people had a mean FICO score of 703 with a standard deviation of 92. Can the economist conclude that the mean fico score is less than 720? Use an alpha level of 0.05.
Reject H0. The evidence supports the claim that the mean fico score is less than 720.
A random sample of 35 Hollywood movies made in the last 10 years had a mean length of 125.2 minutes, with a standard deviation of 20.4 minutes. If a confidence interval was constructed using this data, would it be formed using z, t, or proportion formulas?
T formulas.
A certain soft drink bottler wants to estimate the proportion of its customers that consume their competitors' products on a regular basis. A random sample of 50 customers showed that 15 regularly consume soft drinks from the company's competitors. Construct a 95% confidence interval to estimate the population proportion of interest.
The COO in the company believes that the population proportion is 0.35. Does your confidence interval support this? Why or why not?
(0.173, 0.427)
Yes, because 0.35 is within the confidence interval.
An interior decorator is concerned about recent supply lines for paint colors. She wants to be 98% confident that the company she picks to supply her paint ships within 7 days of the order. How many orders must she examine if the standard deviation is 5.3 days, and she wants a margin of error of .8 days?
238
A poll taken by the software usability research laboratory surveyed 341 video gamers, and 110 of them said that they prefer playing games on a console, rather than on a computer or handheld device. An executive at a game console manufacturing company claims that more than 25% of gamers prefer consoles. Does the poll provide convincing evidence that the claim is true? Use an alpha level of 0.01 in testing your hypothesis.
We Reject H0. We support the claim that more than 25% of gamers prefer consoles.