7.07 Z-scores

Calculate and interpret the z-score for Brock's SAT score.

The formula for finding the z-score is z=\dfrac{x-\mu}{\sigma} where x is a test score, \mu is the mean, and \sigma is the standard deviation.

For the SAT, we are given x=1450,\, \mu=1051,\, and \sigma=211:

Brock's z-score for his SAT test is z=1.89 which means Brock scored 1.89 standard deviations above the mean.

Calculate and interpret the z-score for Brock's ACT score.

We will use the formula for z-scores again, but this time with the given values for the ACT.

For the ACT, we are given x=30,\, \mu=20.8,\, and \sigma=5.7:

Brock's z-score for his ACT test is z=1.61 which means Brock scored 1.61 standard deviations above the mean.

Determine which test Brock did better on relative to all other SAT and ACT test takers.

Compare the z-scores found in parts (a) and (b). The higher Brock's z-score, the better he did relative to the other test takers.

Relative to all people who took the SAT and ACT, Brock did slightly better on his SAT test than he did on his ACT test since his z-score was higher.

Both of Brock's scores were better than average, but similar relative to the averages, so he can report either of the test scores when applying to different colleges. On college applications, only one test score is usually required.

Find the 400\text{ m} sprint time Lina ran during practice.

To find the practice time that had a z-score of -1.27, we can use the formula z=\dfrac{x-\mu}{\sigma} with the given mean, standard deviation, and z-score, then solve for the practice time in seconds.

From the given information, we know \mu=65, \sigma=3, andz=-1.27.

Lina ran a 400\text{ m} practice time of 61.2 seconds.

Running 400\text{ m} in 61.2 seconds is -1.27 standard deviations below the sprinter's average time of 65 seconds. This tells us she ran faster in that practice run than she normally does.

Find the standard deviation of Aurelia's times.

To find the standard deviation, we can use the z-score formula with the given mean, z-score, and practice time, then solve for the standard deviation.

From the given information, we know \mu=62, z=0.85, and x=65.4.

Aurelia's 400\text{ m} times have a standard deviation of 4 seconds.

Find the average 400\text{ m} sprint time for Mariana.

To find the mean, we can use the z-score formula with the given standard deviation, z-score, and practice time, then solve for the mean.

From the given information, we know \sigma=2, z=-0.5, and x=59.5.

Mariana's average 400\text{ m} time is 60.5 seconds.

Of the three sprinters, Mariana has the fastest average 400\text{ m} time, and her sprint times are more consistent.

Describe a method Colette can use to collect data.

First, use Colette's statistical question to determine the type of data that needs to be collected. Then, consider whether the data can be collected by research, a survey, an observation, or a scientific experiment.

Colette needs to collect data on the heights of men and women. Since most people know their heights, Colette can use a survey or poll to collect the data.

Colette could also research the average heights of men and women. While researching, she would need to make sure that the sample is representative of the population, and the data collection process did not introduce bias.

The data Colette collected on the heights of men and women are shown.

Use technology to create a smooth curve to model each distribution and describe the shape of each curve.

Using technology, we can follow these steps to create a smooth curve of the data:

1. Enter the data into a single column using the GeoGebra Statistics calculator.

2. Highlight the data and select One Variable Analysis.

3. In the settings menu (represented by the gear icon), change the frequency type to Normalized. This will adjust the values on the y-axis to reflect a probability distribution.

4. Check the box to show the normal curve. To see the smooth curve on its own, uncheck the histogram box.

First, we will create the smooth curve that approximates the women's heights.

The curve is symmetric and bell-shaped, meaning the data is approximately normal.

Next, we will create the smooth curve that approximates the men's heights.

To see the full curve, we can adjust the settings by selecting the Graph tab and unchecking the automatic dimensions. Then, we can adjust the y-Max to 0.13, which will allow us to see the top of the curve.

Again, the curve is symmetric and bell-shaped, meaning the data is approximately normal.

Although both data sets are normally distributed, they have different measures of center and spread. This means the curves will have a similar shape, but one is likely taller than the other and they are centered around different values.

Answer the statistical question that Colette formulated.

Colette's statistical question was, "How do the heights of men compare to the heights of women?". We can answer this by analyzing the average heights of men and women, which are represented by the center of the normal curves.

Looking at the smooth curves from the previous part, we can see that the curve that approximates the women's heights is centered at around 63 inches. The curve that approximates the men's heights is centered at 70 inches.

This tells us that, on average, men are taller then women.

Rather than using the curves to compare the means, we could have calculated the mean of each data set using technology, then compared the values.

Since both data sets are normally distributed, Colette wanted to further investigate men's and women's heights relative to the height requirement for the ride. Her new statistical question is, "How does the percentage of male riders who can ride this ride compare to the percentage of female riders who can ride?"

Find and interpret the z-scores for the 60-inch height requirement relative to the average American female heights and average American male heights.

The average height of men and women are different, and the standard deviations of the heights are different as well. We can compare the heights of men and women by using z-scores to standardize the measurements.

To find the z-score, we can use the formula z=\dfrac{x-\mu}{\sigma} with the given height requirement. Then, we can use technology to find the mean and standard deviation of each set.

By selecting "Show summary statistics" (\Sigma\text{x} icon), we can find the mean and standard deviation of women's heights.

From this information, we see \mu\approx 62.5 and \sigma\approx 2.5, and we were given x=60.

The z-score for women's height is z=\dfrac{60-62.5}{2.5}=-1.

For women, the height restriction of 60 inches is only 1 standard deviation below the average female height.

Next, we will find the mean and standard deviation of men's heights.

From this information, we see \mu\approx 70, and \sigma\approx 3.

The z-score is z=\dfrac{60-70}{3}\approx -3.33.

The height restriction of 60 inches is more than three standard deviations below the average male height.

Notice that the mean and standard deviations of the sets were rounded to use "nice" values. This allows us to sketch the curves more easily. However, we should not round to "nice" values if the difference is relatively large.

We can use the normal distribution curves of the data to check our answers.

The data value of 60 does lie 1 standard deviation below the mean of the women's heights and 3\frac{1}{3} standard deviations below the mean of the men's heights.

Compare the percentage of male riders who can ride this ride to the percentage of female riders who can ride.

Because we have the z-scores, we can graph the position of the height requirement relative to the men's heights and the women's heights on the same standard normal curve. Then we can use the empirical rule to find and compare the percentages.

Using the empirical rule, we can estimate that more than 99.85\% of men will be able to ride this ride. However, only 84\% of women will be able to ride.