Chapter 7

Point Estimate

-A point estimate is a *single number* that is used to estimate the value of an unknown parameter -The sample mean is a point estimate, because it is a single number -Point estimates are the sample mean

Confidence Level

-Every confidence interval must have a confidence level. -The confidence level is a percentage between 0% and 100% that *measures the success rate of the method used to construct the confidence interval*. -Suppose, we decide to construct a 95% confidence interval for 𝜇. -The *value 95%* is our confidence level. -If we were to draw many samples and use each one to construct a confidence interval, then in the long run, the percentage of confidence intervals that cover the true value would be equal to the confidence level.

Calculating Critical Value

-Find the critical value for 𝑧_(𝛼∕2) for a 92% confidence interval. 1. Find confidence level. The confidence level is 92%, so *1−𝛼 = 0.92*. 2. Find critical value. It follows that *𝛼 = 0.08*, so *𝛼∕2 = 0.04*. The critical value is 𝑧_0.04.  3. Find the z-score that corresponds with the critical value. Since the area to the right of 𝑧_0.04 is 0.04, the area to the left is 1 - 0.04 = 0.96. We find on Table A.2 the critical value to be 1.75. We can also use InvNorm on on the calculator (.96, 0, 1)

Reducing the Margin of Error for Population Proportion

-Sometimes we have a specific value 𝑚 that we would like the margin of error to attain, and we wish to compute a sample size 𝑛 that is likely to give us a margin of error of that size. -Let m represent the margin of error: 𝑚 = 𝑧_(𝛼∕2)x√((p̂(1−p̂))/𝑛) -This formula may be rewritten as: *𝑛=p̂(1−p̂)x[(𝑧_(𝛼∕2)/𝑚)^2]*

Degrees of Freedom Not in the Table

-If the desired number of degrees of freedom isn't listed in Table A.3, then: 1. If the desired number is less than 200, use the next smaller number that is in the table. 2. If the desired number is greater than 200, use the z-value found in the last row of Table A.3, or use Table A.2.

Margin of Error

-If we think that x̅ = 67.30 could be off by as much as 10 points from the population mean, we would estimate μ with the interval 57.30 < μ < 77.30, which could also be written as 67.30 ± 10. -The *plus-or-minus number* is called the margin of error. -We need to determine how large to make the margin of error so that the interval is likely to contain the population mean. -To do this, we use the sampling distribution of x̅.

Confidence Versus Margin of Error

-If we want to be more confident that our interval contains the true value, we must increase the critical value, which increases the margin of error. -There is a trade-off. -We would rather have a higher level of confidence than a lower level, but we would also rather have a smaller margin of error than a larger one.

Interpreting a Confidence Level

-In the cereal boxes example, a 90% confidence interval for the population mean weight 𝜇 was computed to be 20.12 < 𝜇 < 20.38. -It is tempting to say that the *probability* is 90% that 𝜇 is between 20.12 and 20.38. This, however, is not correct. -The term "probability" refers to random events, which can come out differently when experiments are repeated. The numbers 20.12 and 20.38 are fixed, not random. -The population mean is also fixed, even if we do not know precisely what value it is. -The population mean weight is either between 20.12 and 20.38 or it is not. -Therefore we say that *we have 90% confidence that the population mean is in this interval* -On the other hand, let's say that we are discussing a *method* used to construct a 90% confidence interval. -The method will succeed in covering the population mean 90% of the time, and fail the other 10% of the time. -Therefore it is correct to say that *a method for constructing a 90% confidence interval has probability 90% of covering the population mean*

The Student's t-Distribution

-It is very rare that we would know the value of 𝜎 while needing to estimate the value of 𝜇. -In practice, it is more common that 𝜎 is unknown. -When we don't know the value of 𝜎, we replace it with the sample standard deviation s. -However, we cannot then use 𝑧_(𝛼∕2) as the critical value, because the quantity (𝑥−𝜇)/(𝑠∕√𝑛) does not have a normal distribution. -The distribution of this quantity is called the *Student's t distribution*

Determine Which Method to Use

-One of the challenges in constructing a confidence interval is to determine which method to use. The first step is to determine which type of parameter we are estimating. -There are two types of parameters for which we have learned to construct confidence intervals: 1. Population mean 𝜇 2. Population proportion 𝑝

Constructing a Confidence Interval for 𝜇 when 𝜎 is Unknown

-Step 1: Compute the sample mean, x̅, and sample standard deviation, s, if they are not given. -Step 2: Find the number of degrees of freedom n - 1 and the critical value t_(α / 2) -Step 3: Compute the standard error (𝑠∕√𝑛) and multiply it by the critical value to obtain the margin of error t_(a/2)(𝑠∕√𝑛) -Step 4: Use the point estimate and the margin of error to construct the confidence interval: x̅ - t_(a/2) s/√𝑛 < μ < x̅ + t_(a/2) s/√𝑛 -Step 5: Interpret the result

Procedure for Constructing a Confidence Interval for p

-Step 1: Compute the value of the point estimate p̂ = x/n -Step 2: Find the critical value z_(a/2) for the desired confidence level. -Step 3: Compute the standard error √((p̂ (𝟏−p̂))/𝒏) and multiply it by the critical value to obtain the margin of error 𝒛_(𝜶∕𝟐)x√((p̂ (𝟏−p̂ ))/𝒏) -Step 4: Use the point estimate and the margin of error to construct the confidence interval: Point estimate ± Margin of error 𝒑 ± 𝒛_(𝜶∕𝟐)x√((p̂ (𝟏−p̂ ))/𝒏) 𝒑 - 𝒛_(𝜶∕𝟐)x√((p̂ (𝟏−p̂ ))/𝒏) < p < 𝒑 + 𝒛_(𝜶∕𝟐)x√((p̂ (𝟏−p̂ ))/𝒏) -Step 5: Interpret the results

Properties of the Student's t Distribution

-Student's t distributions are *symmetric and unimodal*, just like the normal distribution. -However, they are *more spread out*. -When the number of degrees of freedom is small, this tendency is more pronounced. -When the number of degrees of freedom is large, s tends to be close to 𝜎, so the t distribution is very close to the normal distribution.

Degrees of Freedom

-The Student's t distribution is more spread out than the normal distribution. -The reason is that s is, on the average, a bit smaller than 𝜎. -Also, since s is random, whereas 𝜎 is constant, replacing 𝜎 with s increases the spread. -How much more spread out the distribution is depends on a quantity called the *degrees of freedom*. -The number of degrees of freedom for the Student's t distribution is *1 less than the sample size n*

Confidence Interval for Population Proportion p

-The confidence interval for the population proportion p is: Point estimate ± Margin of error 𝒑 ± 𝒛_(𝜶∕𝟐)x√((p̂ (𝟏−p̂ ))/𝒏)

Necessary Sample Size Formula

-The formula for the required sample size is: *𝑛=p̂(1−p̂ )x[(𝑧_(𝛼∕2)/𝑚)^2] -In order to use this formula, we need a value for 𝑚 and p̂. -We can set the value of 𝑚, but we don't know ahead of time what p̂ is going to be. -There are two ways to determine a value for p̂. 1. Use a value that is available from a previously drawn sample. 2. To assume that p̂ = 0.5, which makes the margin of error as large as possible for any sample size. In this case, the formula simplifies to *𝑛=0.25x[(𝑧_(𝛼∕2)/𝑚)^2]*

Adjusted Sample Proportion 𝑝 ̃

-The method presented for constructing a confidence interval for a proportion requires that we have at least 10 individuals in each category. -When this condition is not met, we can still construct a confidence interval by adjusting the sample proportion a bit. -We increase the number of individuals in each category by 2, so that the sample size increases by 4. -Thus, instead of using the sample proportion p̂=𝑥∕𝑛, we use the *adjusted sample proportion 𝑝 ̃*: *𝑝 ̃=(𝑥+2)/(𝑛+4)*

The Population Proportion and the Sample Proportion

-The population proportion p is unknown. -The sample proportion p̂ is known, and we use the value of p̂ to estimate the unknown value p.

Advantages of the Small-Sample Method

-The small-sample method can be used for any sample size, and recent research has shown that it has two advantages over the traditional method; 1. The margin of error is smaller, because we divide by n + 4 rather than n. 2. The actual probability that the small-sample confidence interval covers the true proportion is almost always at least as great as, or greater than, that of the traditional method.

Standard Error and Critical Values for a Proportion with Small Samples

-The standard error and critical value are calculated in the same way as in the traditional method, except that we use the adjusted sample proportion 𝑝 ̃ in place of p̂, and n + 4 in place of n. -The standard error becomes *√((𝑝 ̃(1−𝑝 ̃))/(𝑛+4))*

Critical Value

-To construct our 95% confidence interval for 𝜇, we begin with a normal curve. -We want to find the z-scores that bound the middle 95% of the area under the curve. -The z-scores are 1.96 and -1.96. -The value 1.96 is called the critical value. -Critical values are the z-scores that bound area we are measuring

The Critical Value 𝑡_(𝛼∕2)

-To find the critical value for a confidence interval, let 1 − 𝛼 be the confidence level expressed as a decimal. -The critical value is then 𝑡_(𝛼∕2), because the area under the Student's t distribution between −𝑡_(𝛼∕2) and 𝑡_(𝛼∕2) is 1 − 𝛼 . -The critical value 𝑡_(𝛼∕2) can be found in Table A.3, in the row corresponding to the number of degrees of freedom and the column corresponding to the desired confidence level; or by technology.

Finding the Critical Value

-We can construct a confidence interval with any confidence level between 0% and 100% by finding the appropriate critical value for that level. -Let *(1-a)* be the confidence level expressed as a decimal. -The critical value is then z_(a/2), because the area under the standard normal curve between -z_(a/2) and z_(a/2), is (1-a).

Sample Size

-We can make the margin of error smaller if we are willing to reduce our level of confidence, but we can also *reduce the margin of error by increasing the sample size*. -If we let m represent the margin of error, then *𝑚 = 𝑧_(𝛼∕2)·𝜎/√𝑛*. -Using algebra, we may rewrite this formula as *𝑛=((𝑧_(𝛼∕2)∙𝜎)/𝑚)^2* which *represents the the sample size needed to achieve the desired margin of error m* -If the value of n given by the formula is not a whole number, round it up to the nearest whole number. -By rounding up we can be sure that the *margin of error is no greater than the desired value m*

Assumptions for Constructing Confidence Interval for p

-We have a simple random sample. -The population is at least 20 times as large as the sample. -The items in the population are divided into two categories. -The sample must contain at least 10 individuals in each category.

How Large to Make the Margin of Error?

-We need to determine how large to make the margin of error so that the interval is likely to contain the population mean. -To do this, we use the sampling distribution of x̅ -When the sample size is large (n > 30), the Central Limit Theorem tells us that the sampling distribution of 𝑥 is approximately normal with mean 𝜇 and standard error 𝜎/√𝑛 .

Rounding off the final result

-When constructing a confidence interval for a population mean, you may be given a value for x̅, or you may be given the data and have to compute x̅ yourself. -If you are given the value of x̅, round the final result to the same number of decimal places as x̅. -If you are given data, then round x̅ and the final result to one more decimal place than is given in the data

Calculating Confidence Interval

1. Find Margin of Error = *Critical value x Standard error* 2. Confidence Interval = *(Point Estimate - Margin of Error)* < 𝜇 < *(Point Estimate + Margin of Error)*

Assumptions for Constructing a Confidence Interval for 𝜇 when 𝜎 is unknown

1. We have a simple random sample. 2. Either the sample size is large (n > 30), or the population is approximately normal. -When the sample size is small ("n"≤30), we must check to determine whether the sample comes from a population that is approximately normal.

Assumptions for Constructing a Confidence Interval for μ When σ Is Known

1. We have a simple random sample. 2. The sample size is large (n > 30), or the population is approximately normal.

70% Confidence Level

Although the Margin of Error is smaller, they cover the population mean only 70% of the time.

Confidence Interval

An *interval* that is used to estimate the value of a parameter.

Procedure for Constructing a Confidence Interval for μ When σ Is Known

Check to be sure the assumptions are satisfied. If they are, then proceed with the following steps. -Step 1: Find the value of the point estimate x − x̅, if it isn't given. -Step 2: Find the critical value z_(a/2) corresponding to the desired confidence level from the last row of Table A.3, from Table A.2, or with technology. -Step 3: Find the standard error 𝜎/√𝑛, and multiply it by the critical value to obtain the *margin of error z_(a/2) 𝜎/√𝑛* -Step 4: Use the point estimate and the margin of error to construct the confidence interval: *Point Estimate ± Margin of Error* x̅ ± z_(a/2) 𝜎/√𝑛 x̅ - z_(a/2) 𝜎/√𝑛 < μ < x̅ + z_(a/2) 𝜎/√𝑛 -Step 5: Interpret the result.

Estimating the Population Mean 𝜇

If you are estimating the population mean 𝝁, there are two methods for constructing a confidence interval: 1. Z-method 2. T-method. -To determine which method to use, we must determine whether 1. the population standard deviation is known, 2. whether the population is approximately normal, and 3. whether the sample size is large (n > 30)

Calculating Margin of Error

Margin of Error = *Critical value x Standard error*

Construct a Confidence Interval for a Population Proportion

NOTATION -p is the population proportion of individuals who are in a specified category. -x is the number of individuals in the sample who are in the specified category. -n is the sample size. -p̂ is the sample proportion of individuals who are in the specified category. *p̂ = x/n*

Margin of Error for Population Proportion

The margin of error is computed as the critical value 𝑧_(𝛼/2) times the standard error: Margin of Error = 𝒛_(𝜶∕𝟐)x√((p̂ (𝟏−p̂ ))/𝒏)

Standard Error

The quantity, *𝜎/√𝑛*, is called the standard error of the mean.

Standard Error for Population Proportion

The standard error is determined by the sampling distribution of 𝑝 ̂ and is given by: Standard Error = √((p̂ (𝟏−p̂))/𝒏)

99.7% Confidence Level

They almost always succeed in covering the population mean, but their margin of error is large.

95% Confidence Level

This represents a good compromise between reliability and margin of error for many purposes.

Margin of Error for Small Samples

𝑧_(𝛼∕2) x √((𝑝 ̃(1−𝑝 ̃))/(𝑛+4))