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 $[tex]$2,200$[/tex] for each claim with a standard deviation of $[tex]$500$[/tex]. 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 $[tex]$2,350$[/tex]. Which most restrictive level of significance would suggest that the company should raise rates?
Upper-Tail Values
a
5 %
2.5 %
1 %
Critical
z-values
1.65
1.96
2.58
A. 1 %
B. 2.5 %
C. 5 %
D. 10 %
Answer (2)
Calculate the z-score: z = 40 500 2350 − 2200 ≈ 1.897 .
Compare the z-score to the critical z-values for different significance levels.
At 5% significance level (critical z-value = 1.65), 1.65"> 1.897 > 1.65 , so reject the null hypothesis.
The most restrictive level of significance is 5% .
Explanation
Understand the problem and provided data We are given the following information:
Population mean (current average cost per claim): μ = $2200
Population standard deviation: σ = $500
Sample size: n = 40
Sample mean: x ˉ = $2350
The statistician believes that the average cost has increased, so we want to determine the most restrictive level of significance that suggests the company should raise rates. We will perform a hypothesis test to determine if the sample mean is significantly greater than the population mean.
Calculate the z-score First, we need to calculate the z-score using the formula: z = n σ x ˉ − μ Plugging in the values, we get: z = 40 500 2350 − 2200 = 40 500 150 z = 500 150 ⋅ 40 z ≈ 500 150 ⋅ 6.3245 ≈ 500 948.68 ≈ 1.897
Compare the z-score to the critical values Now, we compare the calculated z-score (approximately 1.897) to the critical z-values provided in the table for different significance levels:
For a significance level of 5%, the critical z-value is 1.65.
For a significance level of 2.5%, the critical z-value is 1.96.
For a significance level of 1%, the critical z-value is 2.58.
We want to find the most restrictive (smallest) significance level for which the calculated z-score is greater than the critical z-value.
Determine the most restrictive significance level
At a 5% significance level, the critical z-value is 1.65. Since 1.897 > 1.65, we would reject the null hypothesis at this level.
At a 2.5% significance level, the critical z-value is 1.96. Since 1.897 < 1.96, we would fail to reject the null hypothesis at this level.
At a 1% significance level, the critical z-value is 2.58. Since 1.897 < 2.58, we would fail to reject the null hypothesis at this level.
The most restrictive level of significance at which we would reject the null hypothesis (and thus suggest the company should raise rates) is 5%.
Final Answer Therefore, the most restrictive level of significance that suggests the company should raise rates is 5%.
Examples
In business, determining whether to adjust prices often relies on statistical analysis. For example, a store might track the average spending per customer. If a sample of recent transactions shows a significant increase compared to the historical average, the store might consider raising prices. By comparing the calculated z-score to critical values at different significance levels, the store can decide whether the increase is statistically significant enough to warrant a price change. This ensures that pricing decisions are based on solid evidence rather than random fluctuations.
The most restrictive level of significance that suggests Auto Insurance R Us should raise rates is 5%, as the calculated z-score of approximately 1.897 exceeds the critical z-value of 1.65. At this level, we reject the null hypothesis that the average claim cost has not increased. Therefore, based on this analysis, a rate increase may be justified.
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