# 6: Estimating Averages

• 6.1: The Central Limit Theorem for Sums
The central limit theorem tells us that for a population with any distribution, the distribution of the sums for the sample means approaches a normal distribution as the sample size increases. In other words, if the sample size is large enough, the distribution of the sums can be approximated by a normal distribution even if the original population is not normally distributed.
• 6.2: A Single Population Mean using the Student t-Distribution
We rarely know the population standard deviation. In the past, when the sample size was large, this did not present a problem to statisticians. They used the sample standard deviation ss as an estimate for σσ and proceeded as before to calculate a confidence interval with close enough results. However, statisticians ran into problems when the sample size was small. A small sample size caused inaccuracies in the confidence interval.

## 6: Estimating Averages

A manager of a warehouse wants to know how much a typical supplier delivers in 1000 dollar units. He/she takes a sample of 12 suppliers, at random, obtaining the following results:

Supplier Amount Supplier Amount
1 9 7 11
2 8 8 7
3 9 9 13
4 12 10 9
5 9 11 11
6 12 12 10
The computed mean or average of the data = 10. The manager decides to use this as the estimate for expenditure of a typical supplier.

• The "error" = true amount spent minus the estimated amount.
• The "error squared" is the error above, squared.
• The "SSE" is the sum of the squared errors.
• The "MSE" is the mean of the squared errors.

Supplier $Error Error Squared 1 9 -1 1 2 8 -2 4 3 9 -1 1 4 12 2 4 5 9 -1 1 6 12 2 4 7 11 1 1 8 7 -3 9 9 13 3 9 10 9 -1 1 11 11 1 1 12 10 0 0 The SSE = 36 and the MSE = 36/12 = 3. Table of MSE results for example using different estimates So how good was the estimator for the amount spent for each supplier? Let us compare the estimate (10) with the following estimates: 7, 9, and 12. That is, we estimate that each supplier will spend$7, or $9 or$12.

Performing the same calculations we arrive at:

Estimator 7 9 10 12
SSE 144 48 36 84
MSE 12 4 3 7

The estimator with the smallest MSE is the best. It can be shown mathematically that the estimator that minimizes the MSE for a set of random data is the mean. Table showing squared error for the mean for sample data Next we will examine the mean to see how well it predicts net income over time.

The next table gives the income before taxes of a PC manufacturer between 1985 and 1994.

• Good: Liability limits of 50/100/50, $1,000 deductible for collision and comprehensive coverage (full coverage) • Minimum: State minimum liability only, no comprehensive and collision coverage At every coverage level, Nationwide was the cheapest insurance company but GEICO was not very far behind. Continue reading below to see a breakdown of average premiums for each coverage tier. #### Average premiums for "best" full coverage car insurance coverage level We recommend carrying full coverage if you have assets to protect, multiple drivers on your policy (especially teenagers), drive a high-performance or luxury car, or are currently leasing or financing a vehicle. Due to the high liability limits and physical protection provided for your own car at this coverage level, it's typically the most expensive.  Insurance Company 6-Month Premium Monthly Premium Allstate$1,070 $178 Farmers$823 $137 GEICO$611 $102 Liberty Mutual$900 $150 Nationwide$569 $95 Progressive$673 $112 State Farm$690 $115 USAA$672 $112 Not carrying enough liability coverage can leave you at risk of being sued if you cause enough damage to eclipse your liability limits — leaving any assets vulnerable. A$500 deductible is the most common, but you can further decrease your premium by upping your deductible because of the inverse relationship they share — see this illustrated below at the "good" coverage level with a $1,000 deductible. #### Average premiums for "good" full coverage car insurance coverage level We generally recommend keeping your liability limits to at least 50/100/50. This middle-of-the-road level of full coverage also provides comprehensive and collision coverage for your own vehicle with a$1,000 deductible.

 Insurance Company 6-Month Premium Monthly Premium Allstate $877$146 Farmers $687$115 GEICO $503$84 Liberty Mutual $767$128 Nationwide $479$80 Progressive $553$92 State Farm $595$99 USAA $555$92

While a $500 deductible is the most common, you can further decrease your premium with a higher,$1,000 deductible because of the inverse relationship they share. Learn more about how to choose a deductible.

#### Average premiums for "minimum" liability-only car insurance coverage level

Liability limits are set by each state. You must carry at least the state-mandated minimum level of liability insurance in order to be a legal driver in that state. However, keep in mind that this does expose you to more risks:

• A history of having just the minimum level of coverage can reflect negatively on you as a driver in the eyes of an insurance company. They could charge you higher rates because insurers view drivers who consistently carry the minimum amount of insurance as riskier clients.
• In the event of an at-fault accident in which your liability limits aren't sufficient to cover the other driver's injuries and/or property damage, you would be underinsured. You could then be sued to cover the remaining amount.
• If your own vehicle is damaged in an at-fault accident by an uninsured driver or by a comprehensive claim incident (like theft, weather and animal-related damage), you would have no coverage.

Opting for minimum coverage — without comprehensive and collision to cover damage to your own vehicle — is the cheapest tier of auto insurance you can buy.

 Insurance Company 6-Month Premium Monthly Premium Allstate $378$63 Farmers $285$47 GEICO $232$39 Liberty Mutual $315$53 Nationwide $223$37 Progressive $258$43 State Farm $250$42 USAA $233$39

The less coverage you have, the less your premium will cost. However, it's generally recommended to keep your liability coverage as high as possible to ensure your assets are protected. If your vehicle has any considerable value or you're thinking of selling it in the future, make sure you add comprehensive and collision coverage.

### Average car insurance rates by driver age

Age is a major component of auto insurance premiums. Based on traffic safety data, age is a reliable proxy for risk behind the wheel. To help offset the effect of age on auto insurance and find the cheapest possible rates, we recommend comparing car insurance quotes yearly.

See below the relationship between age and car insurance rates.

 Age Group 6-Month Premium Monthly Premium 16-19 $2,286$381 20-29 $943$157 30-39 $724$121 40-49 $697$116 50-59 $649$108 60-69 $658$110 70-79 $762$127 80-85 $882$147

Those aged 50 to 59 pay the least for car insurance, with all other variables constant. Teen drivers pay the most — about $381 per month for drivers between 16 and 19 years old. Once you turn 20, you should expect an average monthly drop in your insurance premium by about$224.

Aside from very young and very old drivers, age doesn't have a major impact on the average cost of auto insurance. Between the ages of 40 and 60, the average difference in premium is only $45. It's important to consider other rating factors that could have a larger impact on premiums. ### Average car insurance premium by driving record Getting any type of violation — even a minor one — can have major impacts on your premium. For an at-fault accident, the average rate increase in the last year was$335 per six-month policy — or $670 per year. Most insurance providers will raise rates for three to five years after any violation, ticket or claim. ### Average cost of car insurance by credit score In all but a handful of states (California, Massachusetts and Hawaii are exceptions), your credit score is a major rating factor. According to the Federal Trade Commission (FTC), drivers with low credit not only file more claims than drivers with high credit but their claims tend to be more expensive. On average, drivers with excellent credit pay$783 less for car insurance than drivers with very poor credit — with all other rating factors constant.

 Credit Tier 6-Month Premium Monthly Premium Very Poor (300-579) $1,424$237 Fair (580-669) $1,127$188 Good (670-739) $930$155 Very Good (740-799) $779$130 Exceptional (800-850) $641$107

### Car insurance rates by gender

In states like California, Hawaii, Massachusetts, Pennsylvania, North Carolina, and Montana, there is no difference in car insurance premiums for men and women. In other states, the difference between car insurance rates for men and women is small — less than a 0.5% difference in car insurance premiums, nationwide.

Although many people assume car insurance costs vary greatly between men and women, it's a minor factor when you look at the bigger picture. However, gender does play a factor in premiums for young drivers. On average, male drivers between the ages of 16 and 19 pay $672 more per year than do female teens. ### Car insurance quote pricing by location Anytime you move, you'll need to update your car insurance. Car insurance is regulated at the state level and priced by ZIP code. Your exact location can have a major impact on your premium. Compare your rates against your state's average to see whether you're paying too much for auto insurance.  State Average 6-Month Premium Monthly Premium Alaska$698 $116 Alabama$648 $108 Arkansas$851 $142 Arizona$708 $118 California$911 $152 Colorado$849 $142 Connecticut$771 $129 Washington DC$713 $119 Delaware$808 $135 Florida$1,162 $194 Georgia$769 $128 Hawaii$540 $90 Iowa$575 $96 Idaho$628 $105 Illinois$642 $107 Indiana$594 $99 Kansas$800 $133 Kentucky$925 $154 Louisiana$1,152 $192 Massachusetts$651 $109 Maryland$718 $120 Maine$550 $92 Michigan$1,267 $211 Minnesota$699 $117 Missouri$844 $141 Mississippi$751 $125 Montana$775 $129 North Carolina$505 $84 North Dakota$661 $110 Nebraska$760 $127 New Hampshire$480 $80 New Jersey$751 $125 New Mexico$637 $106 Nevada$872 $145 New York$846 $141 Ohio$463 $77 Oklahoma$773 $129 Oregon$666 $111 Pennsylvania$669 $112 Rhode Island$937 $156 South Carolina$733 $122 South Dakota$720 $120 Tennessee$629 $105 Texas$749 $125 Utah$657 $110 Virginia$513 $86 Vermont$528 $88 Washington$604 $101 Wisconsin$540 $90 West Virginia$715 $119 Wyoming$730 $122 The difference between the cheapest car insurance state (Ohio) and the most expensive (Michigan) is over$800 per six-month policy period. This means drivers in Michigan pay over $130 per month more for car insurance than do Ohioan drivers! Learn more about auto insurance rates by state. #### State minimum vs. full coverage: state-by-state cost analysis Each state regulates its insurance laws, governing what coverage types drivers must carry and in what amounts. Nearly every state requires certain minimums for liability coverage, while some states may require additional coverage types such as personal injury protection (PIP) or uninsured/underinsured coverage. Below, you'll find a state-by-state rundown of the average six-month premiums for the minimum liability limit versus a "full coverage" policy. While "full coverage" is generally a combination of other coverage types, in this case, it refers to higher liability limits (50/100/50) and collision and comprehensive deductibles at$500 each — a fairly typical coverage level in the United States.

###### AVERAGE 6-MONTH INSURANCE PREMIUMS — FULL COVERAGE VS. STATE MINIMUM
 State State Minimum Liability Full Coverage Alaska $241$698 Alabama $244$648 Arkansas $295$851 Arizona $334$708 California $315$911 Colorado $318$849 Connecticut $360$771 Washington, D.C. $319$713 Delaware $436$808 Florida $539$1,162 Georgia $327$769 Hawaii $211$540 Iowa $150$575 Idaho $199$628 Illinois $238$642 Indiana $221$594 Kansas $248$800 Kentucky $388$925 Louisiana $483$1,152 Massachusetts $266$651 Maryland $333$718 Maine $224$550 Michigan $613$1,267 Minnesota $299$699 Missouri $298$844 Mississippi $283$751 Montana $230$775 North Carolina $201$505 North Dakota $190$661 Nebraska $206$760 New Hampshire $185$480 New Jersey $371$751 New Mexico $252$637 Nevada $437$872 New York $417$846 Ohio $174$463 Oklahoma $296$773 Oregon $320$666 Pennsylvania $226$669 Rhode Island $451$937 South Carolina $323$733 South Dakota $158$720 Tennessee $199$629 Texas $286$749 Utah $320$657 Virginia $197$513 Vermont $166$528 Washington $295$604 Wisconsin $181$540 West Virginia $276$715 Wyoming $167$730

#### Average car insurance rates by region

Car insurance costs vary by region, as well. For example, in the Midwest, Michigan is expensive, but the cheaper car insurance offered in Ohio ($463), Indiana ($594), Illinois ($642) and Wisconsin ($540), help lower the overall average.

###### AVERAGE INSURANCE PREMIUMS BY REGION
New England$619 Great Lakes$697
Great Plains$726 Southwest$737
Rocky Mountains$748 Mideast$753
Southeast$777 Far West$796

### Car insurance quotes by vehicle

It goes without saying: your vehicle contributes to the cost of your car insurance. Every single vehicle will generate a unique premium based on its vehicle identification number (VIN). Insurance companies use the VIN to assess your vehicle’s mileage, accident history and other characteristics that are factored into your premium. It’s difficult to give an average cost of car insurance by vehicle — below are some national averages.

### How car insurance premiums are calculated

An “average rate” is hard to calculate, thanks to the myriad rating factors contributing to any driver's auto insurance premium. For instance, homeowners are likely to pay less for car insurance in general. Your driving record is also a major contributor, as those with a DUI are more likely to pay far higher rates for insurance as well.

Car insurance is designed — and priced — to suit each individual driver, accurately estimating the risk they represent to an insurer. While car insurance quote pricing varies by driver, it also varies by company. Between the cheapest company and the most expensive, you might find a substantial price gap. This is because every insurer weighs different factors when underwriting a policy.

It's important to take the national average cost of car insurance with a grain of salt. The best way to find an affordable policy, consult an insurance agent or compare car insurance online.

### Average car insurance rate FAQs

Across the U.S., the average annual premium is $1,483, or$124 per month. That's a decrease of 4% compared to 2020's national average of $1,548. #### How much does car insurance cost in my state? Your location — down to your specific ZIP code — is a critical rating factor when insurance companies calculate your premium. In addition, because auto insurance is regulated on the state level, each state's laws and regulations do make an impact on average rates. For instance, Michigan is consistently the most expensive state for car insurance largely due to mandated coverage requirements enforced by law. Other expensive states include Louisiana, Florida, Kentucky, Rhode Island, Nevada and California. Some of the cheapest states for car insurance include Ohio, New Hampshire, North Carolina, Virginia and Vermont. #### Which car insurance companies have the most affordable rates? In our survey of top car insurance companies, Nationwide was the cheapest for the average driver profile. GEICO was a close second. Keep in mind that this individual profile likely won't match yours exactly. The best way to find a budget-friendly rate is to shop around for quotes before the end of every policy period. #### How much is car insurance for a young driver in their 20s? On average, drivers aged 20 to 29 pay$1,887 per year for auto insurance. That's $944 for a six-month policy or$157 per month. Young drivers in their 20s paid 27% more than the national average.

The Zebra is not an insurance company. We’re an independent, unbiased partner for consumers, on a mission to help you compare insurance options apples-to-apples, so you can make a truly informed decision. We’re proud because:

• We’re the first to compare all the major insurance companies side-by-side. The information provided on those insurers here is intended to help inform and educate consumers before they decide where to spend their hard-earned cash.
• We have no stake in which insurance company you choose — we simply want to offer you unbiased, fast, side-by-side comparisons of your insurance options.

This article was written by one of The Zebra’s insurance experts . Each article is thoroughly researched to ensure we provide readers the most accurate — and helpful — information possible. That’s insurance in black and white.®

## 6.9: Average Reservoir Pressure Estimating Techniques

Average reservoir pressures are used for characterizing a reservoir, computing its oil/gas in place, and predicting future behavior. In addition to these uses, the average reservoir pressure is required to find a quantitative use in volumetric-balance calculations of oil/gas in place in a reservoir. In this section we will present various methods to calculate average reservoir pressure in a gas reservoir.

### Horner MBH Method

The average reservoir pressure for a finite or bounded reservoir may be estimated as shown below using the values of m and ? />obtained from the Horner plot and the MBH curves.3 From Equation 6 21 for />,

where Eq. 6 22 is the defining equation ? />. The material balance equation may be written in terms of pseudopressure with substitution for dimensionless quantities as

Subtracting Eq. 6 22 from Eq. 6 24 gives

m is the absolute value of the slope of the straight-line section of the Horner plot:

F is the MBH dimensionless pressure at t DA, and the t DA is the dimensionless time:

t P is a pseudoproduction time in hours and is calculated using Eq. 6 17 ? ( p*) is the value of ? ( p ws) corresponding to />, from the extrapolated semilog straight line. F may be obtained from Table B 1 or Figures B 1 through B 5 corresponding to the appropriate well reservoir configuration and reservoir shape. Values of t DA may be calculated from Eq. 6 28.

## Marijuana Law Enforcement Cost States An Estimated $3.6 Billion In 2010: ACLU States together spent somewhere around$3.6 billion enforcing marijuana possession laws in 2010, according to a new study by the American Civil Liberties Union, entitled “The War On Marijuana In Black and White.” That's the authors' "best estimate," though approximations using different methodologies put the cost as high as $6 billion and as low as$1.2 billion.

The paper grabbed headlines Tuesday with its finding that blacks are nearly four times as likely as whites to be arrested for possessing marijuana, despite both races using the drug at about the same rate.

Here are some most startling numbers from the ACLU’s report with regards to the cost of enforcing marijuana laws:

$20 billion: The amount states will spend enforcing marijuana laws over the next six years.$900: The minimum per-capita cost spent by California, Nevada and Washington on criminal justice for marijuana offenders.

$750: The low-level estimate that states pay for each marijuana arrest.$95: The national average per-diem cost of housing an inmate arrested due to a marijuana-related offense.

\$2: The average amount communities spend each day on marijuana supervision.