A nutritionist studying weight gain for college freshmen obtains a sample of n = 20 first year student at the state college. Each student is weighed on the first day of school an again on the last day of the semester. The following scores measure the change in weight, in pounds, for each student. A positive score indicates a weight gain during the semester.
+5, +6, +3, +1,+8, +5, +4,+4, +3, -1
+2, +7, +1, +5, +8, 0, +4, +6, +5, +3
a) Sketch a histogram showing the distribution of weight-change scores.
b) Calculate the mean weight change score for this sample
c) Does there appear to be a consistent trend in weight change during the semester?
Describe the distribution of sample means (shape, expected value, and standard error) for samples of
n _ 36 selected from a population with a mean of ì _ 100 and a standard deviation of _ _ 12.
The distribution of sample means is not always a normal distribution. Under what circumstances is the
distribution of sample means not normal?
For a population with a mean of ì _ 70 and a standard deviation of _ _ 20, how much error, on average,
would you expect between the sample mean (M) and the population mean for each of the following sample
sizes?
a. n _ 4 scores
b. n _ 16 scores
c. n _ 25 scores
If the population standard deviation is _ _ 8, how large a sample is necessary to have a standard error
that is:
a. less than 4 points?
b. less than 2 points?
c. less than 1 point?
For a population with a mean of ì _ 80 and a standard deviation of _ _ 12, find the z-score corresponding to
each of the following samples.
a. M _ 83 for a sample of n _ 4 scores
b. M _ 83 for a sample of n _ 16 scores
c. M _ 83 for a sample of n _ 36 scores
A population forms a normal distribution with a mean of ì _ 80 and a standard deviation of _ _ 15. For
each of the following samples, compute the z-score for the sample mean and determine whether the sample
mean is a typical, representative value or an extreme value for a sample of this size.
a. M _ 84 for n _ 9 scores
b. M _ 84 for n _ 100 scores
The population of IQ scores forms a normal distribution with a mean of ì _ 100 and a standard deviation of _ _ 15. What is the probability of obtaining a sample mean greater than M _ 97,
a. for a random sample of n _ 9 people?
b. for a random sample of n _ 25 people?
A population of scores forms a normal distribution with a mean of ì _ 40 and a standard deviation of _ _ 12.
a. What is the probability of randomly selecting a
score less than X _ 34?
b. What is the probability of selecting a sample of
n _ 9 scores with a mean less than M _ 34?
c. What is the probability of selecting a sample of
n _ 36 scores with a mean less than M _ 34?
At the end of the spring semester, the Dean of Students sent a survey to the entire freshman class. One question
asked the students how much weight they had gained or lost since the beginning of the school year. The average
was a gain of ì _ 9 pounds with a standard deviation of _ _ 6. The distribution of scores was approximately
normal. A sample of n _ 4 students is selected and the average weight change is computed for the sample.
a. What is the probability that the sample mean will
be greater than M _ 10 pounds? In symbols, what
is p(M 10)?
b. Of all of the possible samples, what proportion will
show an average weight loss? In symbols, what is
p(M 0)?
c. What is the probability that the sample mean will
be a gain of between M _ 9 and M _ 12 pounds?
In symbols, what is p(9 M 12)?
The average age for licensed drivers in the county is ì _ 40.3 years with a standard deviation of _ _ 13.2 years.
a. A researcher obtained a random sample of n _ 16
parking tickets and computed an average age of
M _ 38.9 years for the drivers. Compute the z-score
for the sample mean and find the probability of
obtaining an average age this young or younger
for a random sample of licensed drivers. Is it
reasonable to conclude that this set of n _ 16
people is a representative sample of licensed
drivers?
b. The same researcher obtained a random sample of
n _ 36 speeding tickets and computed an average
age of M _ 36.2 years for the drivers. Compute the
z-score for the sample mean and find the probability
of obtaining an average age this young or
younger for a random sample of licensed drivers.
Is it reasonable to conclude that this set of n _ 36
people is a representative sample of licensed drivers?
Welsh, Davis, Burke, and Williams (2002) conducted a study to evaluate the effectiveness of a carbohydrate-electrolyte drink on sports performance and endurance. Experienced athletes were given either a carbohydrate-electrolyte drink or a placebo while they were tested on a series of high-intensity exercises. One measure was how much time it took
for the athletes to run to fatigue. Data similar to the results obtained in the study are shown in the following table.
Time to Run to Fatigue (in minutes) Mean SE Placebo 21.7 2.2 Carbohydrate-electrolyte 28.6 2.7
a. Construct a bar graph that incorporates all of the
information in the table.
b. Looking at your graph, do you think that the
carbohydrate-electrolyte drink helps performance?
a) To sketch a histogram showing the distribution of weight-change scores, follow these steps:
1. Create a horizontal axis labeled "Weight Change (in pounds)".
2. Determine the range of weight change scores. In this case, the scores range from -1 to +8 pounds.
3. Divide the range into equal intervals or bins. Since the range is from -1 to +8, you could divide it into bins of width 1 pound.
4. Count the frequency of each weight change score falling within each bin.
5. Plot a bar for each bin, with the height of the bar representing the frequency of weight change scores within that bin.
b) To calculate the mean weight change score for this sample, follow these steps:
1. Add up all the weight change scores in the sample.
2. Divide the sum by the number of scores in the sample (in this case, 20).
c) To determine if there is a consistent trend in weight change during the semester:
1. Look at the histogram from part (a). Analyze the shape and distribution of the histogram to identify any patterns, such as clusters or skewness.
2. Consider the sign (positive or negative) of the weight change scores. If most of the scores are positive, it suggests a trend of weight gain. If most of the scores are negative, it suggests a trend of weight loss.
3. Calculate any descriptive statistics, such as the median or mode, to further analyze the distribution of weight change scores.
By following these steps, you can sketch a histogram, calculate the mean weight change score, and assess the presence of a consistent trend in weight change during the semester.