Question

car insurance companies want to keep track of the average cost per claim. the current data in use for auto insurance r us is an average of $2,200 for each claim with a standard deviation of $500. with this average, the company can stay competitive with rates but not lose money. however, the statistician for the company believes that the cost of the average claim has increased. he pulled 40 recent claims and found the average to be $2,350. which most restrictive level of significance would suggest that the company should raise rates?

upper-tail values
critical z-values	1.65	1.96	2.58

• 1%
• 2.5%
• 5%
• 10%

Explanation:

Step1: Identify the test type

This is a one - sample z - test for the mean. The formula for the z - statistic in a one - sample z - test is $z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}$, where $\bar{x}$ is the sample mean, $\mu$ is the population mean, $\sigma$ is the population standard deviation, and $n$ is the sample size.

Given: $\mu = 2200$, $\bar{x}=2350$, $\sigma = 500$, $n = 40$.

Step2: Calculate the z - statistic

First, calculate the standard error $SE=\frac{\sigma}{\sqrt{n}}=\frac{500}{\sqrt{40}}\approx\frac{500}{6.3246}\approx79.06$.

Then, calculate the z - statistic: $z=\frac{\bar{x}-\mu}{SE}=\frac{2350 - 2200}{79.06}=\frac{150}{79.06}\approx1.897$.

Step3: Compare with critical values

- For 5% significance level (one - tailed), critical z - value is 1.65. Since $1.897>1.65$, we reject the null hypothesis at 5% level.
- For 2.5% significance level (one - tailed), critical z - value is 1.96. Since $1.897<1.96$, we fail to reject the null hypothesis at 2.5% level.
- For 1% significance level (one - tailed), critical z - value is 2.58. Since $1.897<2.58$, we fail to reject the null hypothesis at 1% level.

The most restrictive level of significance (the smallest $\alpha$) at which we can reject the null hypothesis (and suggest that the company should raise rates) is 5% because at 5% we reject, at 2.5% and 1% we do not. Wait, no, wait. Wait, the null hypothesis is that the average cost per claim has not increased ($\mu\leq2200$), and the alternative is that it has increased ($\mu > 2200$). We calculated $z\approx1.897$.

Critical values:
- 5%: 1.65, $1.897>1.65$ → reject $H_0$
- 2.5%: 1.96, $1.897<1.96$ → fail to reject $H_0$
- 1%: 2.58, $1.897<2.58$ → fail to reject $H_0$

So the most restrictive (the highest $\alpha$ among those where we reject) is 5%? Wait, no, the question is "which most restrictive level of significance would suggest that the company should raise rates". Restrictive here means the smallest $\alpha$ (significance level) at which we can reject the null hypothesis. Wait, 5% is less restrictive than 2.5% and 1%? No, actually, the smaller the $\alpha$, the more restrictive the test. Wait, I think I made a mistake. Let's re - explain:

Significance level $\alpha$: 1% is more restrictive than 2.5% which is more restrictive than 5%. Because a smaller $\alpha$ means we require stronger evidence to reject the null hypothesis.

We have $z = 1.897$.

- At $\alpha = 5\%$ (critical z = 1.65), $1.897>1.65$ → reject $H_0$
- At $\alpha=2.5\%$ (critical z = 1.96), $1.897<1.96$ → fail to reject $H_0$
- At $\alpha = 1\%$ (critical z = 2.58), $1.897<2.58$ → fail to reject $H_0$

So the most restrictive level (the smallest $\alpha$) at which we can reject $H_0$ is 5%? Wait, no, 5% is a larger $\alpha$ than 2.5% and 1%. Wait, the term "most restrictive" here means the level that is the hardest to satisfy (i.e., the smallest $\alpha$) at which we can still reject the null hypothesis. But in this case, we can reject at 5% but not at 2.5% or 1%. So among the options given (1%, 2.5%, 5%, 10%), the most restrictive (the smallest $\alpha$) at which we can reject is 5%? Wait, no, 5% is less restrictive than 2.5% and 1%. I think the confusion is in the term "most restrictive". Actually, when we say a test is more restrictive, it means that it is harder to reject the null hypothesis (smaller $\alpha$). So the question is asking, which is the most restrictive (smallest $\alpha$) level at which we would reject the null hypothesis (and thus suggest the company should raise rates). Since we reject at 5% but not at 2.5% or 1%, the most restrictive level among the options where we reject is 5%? Wait, no, the options are 1%, 2.5%, 5%, 10%. Wait, maybe I miscalculated the z - score.

Wait, let's recalculate the z - score:

$\sigma = 500$, $n = 40$, so standard error $SE=\frac{\sigma}{\sqrt{n}}=\frac{500}{\sqrt{40}}\approx79.0569$.

$\bar{x}-\mu=2350 - 2200 = 150$.

$z=\frac{150}{79.0569}\approx1.897$, which is approximately 1.90.

Critical values:

- 5% one - tailed: 1.645 (or 1.65 as given)
- 2.5% one - tailed: 1.96
- 1% one - tailed: 2.58

So 1.90 > 1.65 (5% level), 1.90 < 1.96 (2.5% level), 1.90 < 2.58 (1% level).

So the null hypothesis is rejected at 5% level, not rejected at 2.5% and 1% levels. So the most restrictive level (the smallest $\alpha$) at which we can reject the null hypothesis is 5%? Wait, no, 5% is a higher $\alpha$ than 2.5% and 1%. The "most restrictive" here should be the highest $\alpha$ among those where we reject? No, the term "most restrictive" in hypothesis testing: a more restrictive test has a smaller $\alpha$, because it's harder to reject the null hypothesis. So if we can reject at 5% but not at 2.5% or 1%, then the most restrictive level (the smallest $\alpha$) at which we can reject is 5%? Wait, that doesn't make sense. Wait, maybe the question is phrased as "which is the most restrictive (i.e., the level that is the least likely to reject, but in this case, we need to find the level at which we can reject, and among those, which is the most restrictive (smallest $\alpha$)". But in the options, 5% is the only one where we reject, except 10% which is not an option? Wait, the options are 1%, 2.5%, 5%, 10%? Wait, the original options are 1%, 2.5%, 5%, 10%? Wait, the user's options are:

- 1%

- 2.5%

- 5%

- 10% (but 10% is not in the critical values table, but the critical value for 10% one - tailed is 1.28, which is less than 1.90, so we would reject at 10% too, but the options given are 1%, 2.5%, 5%, 10%? Wait, no, the user's options are 1%, 2.5%, 5%, 10%? Wait, the image shows options: 1%, 2.5%, 5%, 10%? Wait, the user's question has options: 1%, 2.5%, 5%, 10%? Wait, no, the user's image shows:

Options:

- 1%

- 2.5%

- 5%

- 10%? Wait, no, the user's text says:

"$\bigcirc$ 1%

$\bigcirc$ 2.5%

$\bigcirc$ 5%

$\bigcirc$ 10%"

But the critical values table has 5%, 2.5%, 1%. So maybe 10% is a distractor.

Wait, going back, the z - score is approximately 1.90.

Critical values:

- 5%: 1.65 (reject)

- 2.5%: 1.96 (do not reject)

- 1%: 2.58 (do not reject)

So the null hypothesis is rejected at 5% level, not at 2.5% or 1% levels. So the most restrictive level (the smallest $\alpha$) at which we can reject the null hypothesis is 5%? Wait, no, 5% is a higher $\alpha$ than 2.5% and 1%. The "most restrictive" here is the level that is the hardest to satisfy (smallest $\alpha$) at which we can still reject the null hypothesis. Since we can reject at 5% but not at 2.5% or 1%, then 5% is the most restrictive level among the options where we can reject. Because 5% is smaller than 10% (if 10% was an option, but it's not in the critical values table given, but the options include 10%? Wait, the user's options are 1%, 2.5%, 5%, 10%. But according to the critical values, at 10% one - tailed, critical z is 1.28, so we would reject at 10% too, but the question is about the "most restrictive" level. Since 5% is more restrictive than 10% (because 5% is a smaller $\alpha$ than 10%), and among the options 1%, 2.5%, 5%, 10%, the only one where we reject and is the most restrictive (smallest $\alpha$) is 5%.

Answer:

5% (the option with 5%)