Math / Data & Statistics

QuestionUnhealthy Days in Cities The number of unhealthy days based on the AQI (Air Quality Index) for a random sample of metropolitan areas is shown. Round the sample statistics and final answers to at least one decimal place. $\begin{array}{lllllllll}29 & 23 & 10 & 32 & 24 & 0 & 5 & 27\end{array}$ Send data to Excel
Construct a $98 \%$ confidence interval based on the data. Assume the variable is normally distributed. $8.49^{8}<\mu<35.01^{8}$

Studdy Solution

STEP 1

1. The data represents the number of unhealthy days based on the AQI for a sample of metropolitan areas.
2. The variable is normally distributed.
3. We are constructing a $98\%$ confidence interval for the mean.
4. The sample size is $n = 8$ .

STEP 2

1. Calculate the sample mean ( $\bar{x}$ ).
2. Calculate the sample standard deviation ( $s$ ).
3. Determine the critical value ( $t^*$ ) for a $98\%$ confidence level.
4. Calculate the margin of error.
5. Construct the confidence interval.

STEP 3

Calculate the sample mean ( $\bar{x}$ ):
$\bar{x} = \frac{29 + 23 + 10 + 32 + 24 + 0 + 5 + 27}{8}$
$\bar{x} = \frac{150}{8} = 18.75$

STEP 4

Calculate the sample standard deviation ( $s$ ):
First, find the deviations from the mean and square them:
$(29 - 18.75)^2, (23 - 18.75)^2, (10 - 18.75)^2, (32 - 18.75)^2, (24 - 18.75)^2, (0 - 18.75)^2, (5 - 18.75)^2, (27 - 18.75)^2$
Calculate each squared deviation:
$104.0625, 18.0625, 76.5625, 174.0625, 27.5625, 351.5625, 189.0625, 67.5625$
Sum the squared deviations and divide by $n-1$ :
$s^2 = \frac{104.0625 + 18.0625 + 76.5625 + 174.0625 + 27.5625 + 351.5625 + 189.0625 + 67.5625}{7}$
$s^2 = \frac{1008.5}{7} = 144.0714$
Calculate the standard deviation:
$s = \sqrt{144.0714} \approx 12.0$

STEP 5

Determine the critical value ( $t^$ ) for a $98\%$ confidence level with $n-1 = 7$ degrees of freedom. Using a $t$ -distribution table or calculator, find $t^ \approx 2.998$ .

STEP 6

Calculate the margin of error:
$\text{Margin of Error} = t^* \times \frac{s}{\sqrt{n}}$
$\text{Margin of Error} = 2.998 \times \frac{12.0}{\sqrt{8}}$
$\text{Margin of Error} \approx 2.998 \times 4.24 \approx 12.7$

STEP 7

Construct the confidence interval:
$\bar{x} - \text{Margin of Error} < \mu < \bar{x} + \text{Margin of Error}$
$18.75 - 12.7 < \mu < 18.75 + 12.7$
$6.05 < \mu < 31.45$
The $98\%$ confidence interval for the mean number of unhealthy days is:
$\boxed{6.05 < \mu < 31.45}$

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Studdy Solution

STEP 1

1. The data represents the number of unhealthy days based on the AQI for a sample of metropolitan areas.
2. The variable is normally distributed.
3. We are constructing a $98\%$ confidence interval for the mean.
4. The sample size is $n = 8$ .

STEP 2

1. Calculate the sample mean ( $\bar{x}$ ).
2. Calculate the sample standard deviation ( $s$ ).
3. Determine the critical value ( $t^*$ ) for a $98\%$ confidence level.
4. Calculate the margin of error.
5. Construct the confidence interval.

STEP 3

Calculate the sample mean ( $\bar{x}$ ):
$\bar{x} = \frac{29 + 23 + 10 + 32 + 24 + 0 + 5 + 27}{8}$
$\bar{x} = \frac{150}{8} = 18.75$

STEP 4

STEP 5

Determine the critical value ( $t^$ ) for a $98\%$ confidence level with $n-1 = 7$ degrees of freedom. Using a $t$ -distribution table or calculator, find $t^ \approx 2.998$ .

STEP 6

Calculate the margin of error:
$\text{Margin of Error} = t^* \times \frac{s}{\sqrt{n}}$
$\text{Margin of Error} = 2.998 \times \frac{12.0}{\sqrt{8}}$
$\text{Margin of Error} \approx 2.998 \times 4.24 \approx 12.7$

STEP 7

Construct the confidence interval:
$\bar{x} - \text{Margin of Error} < \mu < \bar{x} + \text{Margin of Error}$
$18.75 - 12.7 < \mu < 18.75 + 12.7$
$6.05 < \mu < 31.45$
The $98\%$ confidence interval for the mean number of unhealthy days is:
$\boxed{6.05 < \mu < 31.45}$

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Studdy Solution

STEP 1

1. The data represents the number of unhealthy days based on the AQI for a sample of metropolitan areas.2. The variable is normally distributed.3. We are constructing a 98%98\%98% confidence interval for the mean.4. The sample size is n=8n = 8n=8.

STEP 2

1. Calculate the sample mean (xˉ\bar{x}xˉ).2. Calculate the sample standard deviation (sss).3. Determine the critical value (t∗t^*t∗) for a 98%98\%98% confidence level.4. Calculate the margin of error.5. Construct the confidence interval.

STEP 3

Calculate the sample mean (xˉ\bar{x}xˉ):xˉ=29+23+10+32+24+0+5+278\bar{x} = \frac{29 + 23 + 10 + 32 + 24 + 0 + 5 + 27}{8}xˉ=829+23+10+32+24+0+5+27​xˉ=1508=18.75\bar{x} = \frac{150}{8} = 18.75xˉ=8150​=18.75

STEP 4

STEP 5

Determine the critical value (t∗t^*t∗) for a 98%98\%98% confidence level with n−1=7n-1 = 7n−1=7 degrees of freedom. Using a ttt-distribution table or calculator, find t∗≈2.998t^* \approx 2.998t∗≈2.998.

STEP 6

STEP 7

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1. The data represents the number of unhealthy days based on the AQI for a sample of metropolitan areas.
2. The variable is normally distributed.
3. We are constructing a $98\%$ confidence interval for the mean.
4. The sample size is $n = 8$ .

1. Calculate the sample mean ( $\bar{x}$ ).
2. Calculate the sample standard deviation ( $s$ ).
3. Determine the critical value ( $t^*$ ) for a $98\%$ confidence level.
4. Calculate the margin of error.
5. Construct the confidence interval.

Calculate the sample mean ( $\bar{x}$ ):
$\bar{x} = \frac{29 + 23 + 10 + 32 + 24 + 0 + 5 + 27}{8}$
$\bar{x} = \frac{150}{8} = 18.75$

Determine the critical value ( $t^$ ) for a $98\%$ confidence level with $n-1 = 7$ degrees of freedom. Using a $t$ -distribution table or calculator, find $t^ \approx 2.998$ .