Mathematics, 10.06.2020 04:57 carley608

Activity trackers are electronic devices that people wear to record physical activity. Researchers wanted to estimate the mean number of steps taken on a typical workday for people working in New York City who wear such trackers. A random sample of 61 people working in New York City who wear an activity tracker was selected. The number of steps taken on a typical workday for each person in the sample was recorded. The mean was 9,797 steps and the standard deviation was 2,313 steps. a. Construct and interpret a 99 percent confidence interval for the mean number of steps taken on a typical workday for all people working in New York City who wear an activity tracker.
b. A wellness director at a company in New York City wants to investigate whether it is unusual for one person working in the city who wears an activity tracker to record approximately 8,500 steps on a typical workday. Is it appropriate to use the confidence interval found in part (a) to conduct the investigation.

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a) The 99% confidence interval for the mean number of steps taken on a typical workday for all people working in New York City who wear an activity tracker is (9,009, 10,585).

We are 95% confident that the mean number of steps taken on a typical workday for all people working in New York City who wear an activity tracker is within 9,009 and 10,585 steps.

b) No, we can not use the confidence interval to estimate the probability of individual values. It can onlybe used to make inference about the population mean.

Step-by-step explanation:

a) We have to calculate a 99% confidence interval for the mean.

The population standard deviation is not known, so we have to estimate it from the sample standard deviation and use a t-students distribution to calculate the critical value.

The sample mean is M=9,797.

Ths sample standard deviation is s=2,313.

The sample size is N=61.

When σ is not known, s divided by the square root of N is used as an estimate of σM:

$s_M=\dfrac{s}{\sqrt{N}}=\dfrac{2313}{\sqrt{61}}=\dfrac{2313}{7.81}=296.15$

The degrees of freedom for this sample size are:

df=n-1=61-1=60

The t-value for a 99% confidence interval and 61 degrees of freedom is t=2.66.

The margin of error (MOE) can be calculated as:

$MOE=t\cdot s_M=2.66 \cdot 296.15=787.84$

Then, the lower and upper bounds of the confidence interval are:

$LL=M-t \cdot s_M = 9797-787.84=9009\\\\UL=M+t \cdot s_M = 9797+787.84=10585$

The 99% confidence interval for the mean number of steps taken on a typical workday for all people working in New York City who wear an activity tracker is (9,009, 10,585).

b) The value of 8,500 steps is outside the confidence interval, but this means that it is an unusual value for the mean number of steps for all people in New York City who wear an activity tracker.

We can not use the confidence interval to estimate the probability of individual values.

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Activity trackers are electronic devices that people wear to record physical activity. Researchers w...