2. an athletic trainer at the university of missouri hypothesizes that football players drink more water than soccer players during the season, so he sends out a survey to test his hypothesis. the athletes were asked to record how many ounces of water they consumed over the course of one full day. the data for the athletes is below:

| | average ounces of h₂o consumed ± 2seₓ |
| --- | --- |
| 50 athletes surveyed in football | 70 ± 13 |
| 50 athletes surveyed in soccer | 64 ± 10 |

a. state a null hypothesis for this experiment.

b. use the space provided to make a bar graph comparing the sample means given in the table above. make sure to plot the 95% ci (+/- 2sem), using the values provided.

university of missouri athlete
consumption of water study

graph grid with x - axis: football, soccer; y - axis: mean ounces of h₂o consumed

c. what conclusion can be drawn from this data? circle the correct options below to complete the conclusion:

- there is / is not sufficient evidence to reject the null hypothesis. the athletic trainer’s alternative hypothesis is / is not supported by the data. while the mean ounces of water consumed by football players is higher than the mean ounces of water consumed by soccer players, the error bars overlap / do not overlap indicating that the difference between the means is / is not significant.

3. you are comparing three data sets, each with the same standard deviation of 2.3. the only difference between the data sets is the sample size (n). data set 1 has a sample size of 15, data set 2 has a sample size of 30, and data set 3 has a sample size of 45. calculate the standard error for each data set.

a. data set 1: ____________

b. data set 2: ____________

c. data set 3: ____________

4. referring to question 3, what happened to the standard error when the sample size (n) was increased? use the formula to justify why this happened.

Question

2. an athletic trainer at the university of missouri hypothesizes that football players drink more water than soccer players during the season, so he sends out a survey to test his hypothesis. the athletes were asked to record how many ounces of water they consumed over the course of one full day. the data for the athletes is below:

| | average ounces of h₂o consumed ± 2seₓ |
| --- | --- |
| 50 athletes surveyed in football | 70 ± 13 |
| 50 athletes surveyed in soccer | 64 ± 10 |

a. state a null hypothesis for this experiment.

b. use the space provided to make a bar graph comparing the sample means given in the table above. make sure to plot the 95% ci (+/- 2sem), using the values provided.

university of missouri athlete
consumption of water study

graph grid with x - axis: football, soccer; y - axis: mean ounces of h₂o consumed

c. what conclusion can be drawn from this data? circle the correct options below to complete the conclusion:

- there is / is not sufficient evidence to reject the null hypothesis. the athletic trainer’s alternative hypothesis is / is not supported by the data. while the mean ounces of water consumed by football players is higher than the mean ounces of water consumed by soccer players, the error bars overlap / do not overlap indicating that the difference between the means is / is not significant.

3. you are comparing three data sets, each with the same standard deviation of 2.3. the only difference between the data sets is the sample size (n). data set 1 has a sample size of 15, data set 2 has a sample size of 30, and data set 3 has a sample size of 45. calculate the standard error for each data set.

a. data set 1: ____________

b. data set 2: ____________

c. data set 3: ____________

4. referring to question 3, what happened to the standard error when the sample size (n) was increased? use the formula to justify why this happened.

Accepted Answer

### Question 2a
# Brief Explanations:
The null hypothesis assumes no difference between the groups. Here, it's that football and soccer players drink the same amount of water.
# Answer:
The null hypothesis ($H_0$) is: There is no difference in the average ounces of water consumed between football players and soccer players (i.e., the average ounces of water consumed by football players is equal to the average ounces of water consumed by soccer players).

### Question 2b
# Brief Explanations:
1. **Axes Setup**: The x - axis has two categories: Football and Soccer. The y - axis is labeled "Mean Ounces of $H_2O$ Consumed".
2. **Bar Heights**: For Football, the mean is 70, so the bar height is 70. For Soccer, the mean is 64, so the bar height is 64.
3. **Error Bars**:
   - For Football, the error is $\pm13$. So the lower bound is $70 - 13=57$ and the upper bound is $70 + 13 = 83$.
   - For Soccer, the error is $\pm10$. So the lower bound is $64 - 10 = 54$ and the upper bound is $64+10 = 74$.
   - Draw vertical error bars on each bar from the lower bound to the upper bound.

(Note: Since this is a text - based response, we can't draw the actual graph, but this is the process to create it.)

### Question 2c
# Brief Explanations:
1. **Error Bar Overlap**: The confidence interval for Football is $57 - 83$ and for Soccer is $54 - 74$. These intervals overlap.
2. **Null Hypothesis and Alternative Hypothesis**: When error bars overlap, we do not have sufficient evidence to reject the null hypothesis, and the alternative hypothesis (that football players drink more) is not supported. The difference between the means is not significant because of the overlapping error bars.
# Answer:
There $\boldsymbol{\text{is not}}$ sufficient evidence to reject the null hypothesis. The athletic trainer's alternative hypothesis $\boldsymbol{\text{is not}}$ supported by the data. While the mean ounces of water consumed by football players is higher than the mean ounces of water consumed by soccer players, the error bars $\boldsymbol{\text{overlap}}$ indicating that the difference between the means $\boldsymbol{\text{is not}}$ significant.

### Question 3
The formula for standard error ($SE$) is $SE=\frac{\sigma}{\sqrt{n}}$, where $\sigma$ is the standard deviation and $n$ is the sample size. Given $\sigma = 2.3$.

#### 3a. Data set 1 ($n = 15$)
# Explanation:
## Step 1: Recall the standard error formula
The formula for standard error is $SE=\frac{\sigma}{\sqrt{n}}$, where $\sigma = 2.3$ and $n = 15$.
## Step 2: Substitute the values into the formula
$SE=\frac{2.3}{\sqrt{15}}$
First, calculate $\sqrt{15}\approx3.873$
Then, $SE=\frac{2.3}{3.873}\approx0.594$
# Answer:
$\approx0.594$

#### 3b. Data set 2 ($n = 30$)
# Explanation:
## Step 1: Recall the standard error formula
$SE=\frac{\sigma}{\sqrt{n}}$, with $\sigma = 2.3$ and $n = 30$.
## Step 2: Substitute the values into the formula
$SE=\frac{2.3}{\sqrt{30}}$
Calculate $\sqrt{30}\approx5.477$
Then, $SE=\frac{2.3}{5.477}\approx0.420$
# Answer:
$\approx0.420$

#### 3c. Data set 3 ($n = 45$)
# Explanation:
## Step 1: Recall the standard error formula
$SE=\frac{\sigma}{\sqrt{n}}$, where $\sigma = 2.3$ and $n = 45$.
## Step 2: Substitute the values into the formula
$SE=\frac{2.3}{\sqrt{45}}$
Calculate $\sqrt{45}\approx6.708$
Then, $SE=\frac{2.3}{6.708}\approx0.343$
# Answer:
$\approx0.343$

### Question 4
# Brief Explanations:
1. **Formula Analysis**: The formula for standard error is $SE=\frac{\sigma}{\sqrt{n}}$. The standard deviation ($\sigma$) is constant (2.3 in this case). The sample size ($n$) is in the denominator under a square root.
2. **Effect of Increasing $n$**: As $n$ increases, $\sqrt{n}$ increases. Since $SE$ is inversely proportional to $\sqrt{n}$ (when $\sigma$ is constant), as $n$ increases, $SE$ decreases. For example, when $n$ went from 15 to 30 to 45, the standard error decreased from approximately 0.594 to 0.420 to 0.343.
# Answer:
When the sample size ($n$) was increased, the standard error ($SE$) decreased. This is justified by the formula for standard error, $SE=\frac{\sigma}{\sqrt{n}}$. Since the standard deviation ($\sigma$) is constant, as $n$ (the sample size) increases, the value of $\sqrt{n}$ increases. Because $SE$ is inversely proportional to $\sqrt{n}$ (when $\sigma$ is held constant), an increase in $n$ leads to a decrease in $SE$.

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QUESTION IMAGE

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Explanation:

Question 2a

Question 2c

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Question 2b