A scientist estimates that the mean nitrogen dioxide level in a city is greater than 32 parts per billion. To test this estimate, you determine the nitrogen dioxide $$ \quad $$ 39 43 levels for 31 randomly selected days. The results (in parts per billion) are listed to the right. Assume that the population standard deviation is $$ 7 . A t \boldsymbol{\alpha}=0.05 $$, can 173716343522303334 you support the scientist's estimate? Complete parts (a) through (e). (a) Write the claim mathematically and identify $$ \mathrm{H}_{0} $$ and $$ \mathrm{H}_{a} $$. Choose from the following. A. $$ H_{0} \cdot \mu=32 $$ B $$ \mathrm{H}_{\mathrm{a}} . \mu32 \text { (claim) } $$ $$ \begin{array}{l} H_{0} \mu \leq 32 \ H_{a}: \mu32 \text { (claim) } \end{array} $$ D. $$ \mathrm{H}_{0}-\mu32 $$ F. $$ \mathrm{H}_{0}: \mu \geq 32 $$ (claim) $$ \mathrm{H}_{\mathrm{a}} \cdot \mu

Question

A scientist estimates that the mean nitrogen dioxide level in a city is greater than 32 parts per billion. To test this estimate, you determine the nitrogen dioxide $$ \quad $$ 39 43 levels for 31 randomly selected days. The results (in parts per billion) are listed to the right. Assume that the population standard deviation is $$ 7 . A t \boldsymbol{\alpha}=0.05 $$, can 173716343522303334 you support the scientist's estimate? Complete parts (a) through (e). (a) Write the claim mathematically and identify $$ \mathrm{H}_{0} $$ and $$ \mathrm{H}_{a} $$. Choose from the following. A. $$ H_{0} \cdot \mu=32 $$ B $$ \mathrm{H}_{\mathrm{a}} . \mu>32 \text { (claim) } $$ $$ \begin{array}{l} H_{0} \mu \leq 32 \ H_{a}: \mu>32 \text { (claim) } \end{array} $$ D. $$ \mathrm{H}_{0}-\mu<32 $$ E $$ \mathrm{H}_{0}-\mu \leq 32 $$ (claim) $$ \mathrm{H}_{\mathrm{a}}-\mu \geq 32 $$ (claim) $$ \mathrm{H}_{\mathrm{a}} \cdot \mu=32 $$ c. $$ H_{0}: \mu=32 $$ (claim) $$ \mathrm{H}_{\mathrm{a}}: \mu>32 $$ F. $$ \mathrm{H}_{0}: \mu \geq 32 $$ (claim) $$ \mathrm{H}_{\mathrm{a}} \cdot \mu<32 $$ (b) Find the critical value and identify the rejection region. $$ z_{0}=1.65 $$ (Round to two decimal places as needed.) Rejection region: z $$ \square $$ 1.65 (c) Find the standardized test statistic. $$ \mathrm{z}= $$ $$ \square $$ (Round to two decimal places as needed.)

UpStudy AI · Accepted Answer

To solve this problem, we will follow the steps outlined in the prompt.

### (a) Write the claim mathematically and identify $$ H_0 $$ and $$ H_a $$.

The scientist's claim is that the mean nitrogen dioxide level in the city is greater than 32 parts per billion. This can be expressed mathematically as:

- Null Hypothesis ($$ H_0 $$): The mean nitrogen dioxide level is less than or equal to 32 parts per billion.
- Alternative Hypothesis ($$ H_a $$): The mean nitrogen dioxide level is greater than 32 parts per billion.

Thus, we can write:

$$
H_0: \mu \leq 32
$$
$$
H_a: \mu > 32 \text{ (claim)}
$$

The correct choice from the options provided is:
- **B.**
$$
H_0: \mu \leq 32
$$
$$
H_a: \mu > 32 \text{ (claim)}
$$

### (b) Find the critical value and identify the rejection region.

Given that this is a one-tailed test with $$ \alpha = 0.05 $$, we will find the critical value $$ z_0 $$ for a right-tailed test.

The critical value for $$ z $$ at $$ \alpha = 0.05 $$ is approximately $$ 1.645 $$ (rounded to two decimal places, it is $$ 1.65 $$).

Thus, the rejection region is:
$$
z > 1.65
$$

### (c) Find the standardized test statistic.

To find the standardized test statistic $$ z $$, we will use the formula:

$$
z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}
$$

Where:
- $$ \bar{x} $$ = sample mean
- $$ \mu_0 $$ = hypothesized mean (32)
- $$ \sigma $$ = population standard deviation (7)
- $$ n $$ = sample size (31)

First, we need to calculate the sample mean $$ \bar{x} $$ from the provided nitrogen dioxide levels. Let's assume the nitrogen dioxide levels for the 31 days are as follows (you will need to provide the actual data for accurate calculations):

Assuming the nitrogen dioxide levels are:
$$ 39, 43, \ldots $$ (31 values)

Let's calculate the sample mean $$ \bar{x} $$ and then use it to find the standardized test statistic $$ z $$.

1. Calculate the sample mean $$ \bar{x} $$:
   $$
   \bar{x} = \frac{\text{sum of all nitrogen dioxide levels}}{n}
   $$

2. Substitute $$ \bar{x} $$, $$ \mu_0 $$, $$ \sigma $$, and $$ n $$ into the formula for $$ z $$.

Please provide the complete list of nitrogen dioxide levels so I can calculate $$ \bar{x} $$ and subsequently the standardized test statistic $$ z $$.
(a) $$ H_0: \mu \leq 32 $$, $$ H_a: \mu > 32 $$ (claim) (b) Rejection region: $$ z > 1.65 $$ (c) $$ z = $$ (calculate using sample mean)

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