T Intervals for Means

A sample of 20 cupcakes found the interval for average calories to be (150, 350). Which is the correct interpretation of the 95% confidence interval?

• We are 95% confident that the true mean caloric content can be found with a sample of 150 to 350 cupcakes.

• We are 95% confident that the interval (150, 350) captures the true average caloric content.

• We are 95% confident that a sample of 20 cupcakes will find 250 calories per cupcake.

• None of these are correct.

Thirty randomly selected students took the calculus final. If the sample mean was 92 and the standard deviation was 9.4, construct a 99 percent confidence interval for the mean score of all students from which the sample was gathered.

• 89.08 < μ < 94.92

• 87.27 < μ < 96.73

• 87.29 < μ < 96.71

• 87.77 < μ < 96.23

The lifetime of a certain type of batter is known to be normally distributed with a standard deviation of 20 hours. A sample of 50 batteries had a mean lifetime of 120.1 hours. It is desired to construct a 99% confidence interval for the mean lifetime of this type of battery. What is the Confidence Interval to one decimal place?

• (114.5, 125.6)

• (112.8, 127.4)

• (112.82, 127.38)

• (114.56, 125.64)

A quality control specialist at a glass factory must estimate the mean clarity rating for a new batch of glass using a sample of 18 glass sheets from the batch. Past investigations show that clarity ratings are normally distributed. The specialist decides to use a t-distribution rather than a z-distribution because ...

• The t distribution is more accurate than a z distribution.

• Clarity ratings for the entire bacth are normally distributed.

• The t-distribution will create a narrower interval than z.

• The standard deviation for the population is unknown.

Why do we use a t-distribution instead of a z-distribution for means?

• Because the population standard deviation is UNKNOWN and it accounts for a greater variability present in SMALL samples.

• It allows us to decrease the margin of error without increasing our confidence level.

• It allows us to use the population standard deviation.

• Because the population standard deviation is KNOWN and it accounts for a greater variability present in LARGE samples.

Researchers suspect that Variety A tomato plants have a different average yield than Variety B tomato plants. To find out, researchers randomly select 10 Variety A and 10 Variety B tomato plants. Then the researchers divide in half each of 10 small plots of land in different locations. For each plot, a coin toss determines which half of the plot gets a Variety A plant; a Variety B plant goes in the other half. After harvest, they compare the yield in pounds for the plants at each location. The 10 differences (Variety A - Variety B) in yield are recorded. A graph of the differences looks roughly symmetric and unimodal with no outliers. Which of the following is the best reason to use a one-sample t interval for a mean difference rather than a two-sample t interval for a difference in means to analyze these data?

• The number of plots is the same for Variety A and Variety B plants.

• The response variable, yield of tomatoes, is quantitative.

• This is an experiment with randomly assigned treatments.

• Each plot is given both varieties of tomato plant.

• The sample size is less than 30 for both treatments.

One reason for using a t distribution instead of the standard Normal distribution to find critical values when calculating a level C confidence interval for a population mean is that...

• z can be used only for large samples.

• z requires that you know the population standard deviation σ.

• z requires that you can regard your data as an SRS from the population.

• z requires that the sample size is less than 10% of the en population size.

• a z critical value will lead to a wider interval than at critical value.

Calculate the mean of the following interval level data: 10, 15, 20, 25, 30

• 35

• 15

• 25

• 20

Find the mean, for the given distribution:

• 4.2

• 2.1

• 4

• 2

Answer explanation

Which of the following does NOT have an affect on the width of a confidence interval?

• Sample mean

• t*

• Sample standard deviation

• Sample size