stats exam 3

A potential candidate for president has stated that she will run for office if at least 30% of Americans voice support for her candidacy. To make her decision, she draws a random sample of 500 Americans. Suppose that in fact 35% of all Americans support her candidacy. The mean for the sampling distribution of is 0.30 0.35 Cannot be determined without more information.

0.35

Many people believe that playing chess can have a positive effect on reading ability. A sample of n = 8 chess players was taken. After participating in a comprehensive chess program, each individual was given a post-test to gauge reading ability. If we want to create a 95% confidence interval for the true mean score, what is the correct t* for this problem?

2.365

Common mistake!!!

A common mistake is to calculate a one-sample confidence interval for μ1 and then check whether μ2 falls within that confidence interval, or vice versa. This is WRONG because the variability in the sampling distribution for two independent samples is more complex and must take into account variability coming from both samples—hence the more complex formula for the standard error.

"Plus four" confidence interval for p

A simple adjustment produces more accurate confidence intervals. We act as if we had four additional observations, two being successes and two being failures. Thus, the new sample size is n + 4 and the count of successes is X + 2. The "plus four" estimate of p is: And an approximate level C confidence interval is: Use this method when C is at least 90% and sample size is at least 10.

Conditions for inference on p

Assumptions: 1. We regard our data as a simple random sample (SRS) from the population. That is, as usual, the most important condition. 2. The sample size n is large enough that the sampling distribution is indeed normal. How large a sample size is enough? Different inference procedures require different answers (we'll see what to do practically).

Two sample t-confidence interval

Because we have two independent samples we use the difference between both sample averages ( 1 − 2) to estimate (μ1 − μ2). Practical use of t: t* C is the area between −t* and t*. We can use the conservative df and Minitab to find t*, or let Minitab create the entire confidence interval for the exact df. The margin of error m is:

When we replace σ with s in our standardized test statistic formula, the distribution of the test statistic

Changes to a t-distribution.

Large-sample confidence interval for p

Confidence intervals contain the population proportion p in C% of samples. For an SRS of size n drawn from a large population and with sample proportion hatp calculated from the data, an approximate level C confidence interval for p is:

If you increase the acceptable margin of error from 0.2 to 0.3, the required sample size will Decrease. Remain the same. Increase. Either increase or decrease because the sample sizes vary according to chance.

Decrease.

A group of psychologists was interested in knowing if the living environment had any effect on a student's GPA. They took a set of twins and randomly assigned one twin to live in an urban area and the other twin to live in a rural area. After one year, they computed the GPAs for the twins and looked at the differences. The P-value was found to be between 0.20 and 0.25. What can you conclude if α = 0.05? 1. Reject H0 and say that a difference exists between the mean GPAs. 2. Reject H0 and say there is insufficient evidence to say that a difference exists between the mean GPAs. 3. Do not reject H0 and say that a difference exists between the mean GPAs. 4. Do not reject H0 and say there is insufficient evidence to say that a difference exists between the mean GPAs.

Do not reject H0 and say there is insufficient evidence to say that a difference exists between the mean GPAs.

The theoretical sampling distribution of hatp Gives the values of hatp from all possible samples of size n from the same population. Provides information about the shape, center, and spread of the values in a single sample. Can only be constructed from the results of a single random sample of size n. Is another name for the histogram of the values in a random sample of size n.

Gives the values of hatp from all possible samples of size n from the same population.

A group of psychologists was interested in knowing if the living environment had any effect on a student's GPA. They took a set of twins and randomly assigned one twin to live in an urban area and the other twin to live in a rural area. After one year, they computed the GPAs for the twins and looked at the differences. What are the hypotheses of interest?

H0: ud= vs Ha: Ud = 0

An experiment was conducted to see if elderly patients had more trouble keeping their balance when loud, unpredictable noises were made compared to younger patients who were also exposed to the noises. Researchers compared the amount of forward and backward sway for the two groups. If we wanted to test whether the younger patients had less average forward/backward sway, we would use which of the following hypotheses?

Ho: u/elderly= u/young versus Ha: u/elderly > u/young

A tire manufacturer claims that one particular type of tire will last at least 50,000 miles. A group of angry customers does not believe this is so. They take a sample of 14 tires and want to test if the mean mileage of the tires is really less than 50,000. What set of hypotheses are they interested in testing?

Ho: u=50000 vs Ha: u < 50,000

The standard error

Is an estimate, using sample data, of the standard deviation of the sampling distribution of hatp .

A group of psychologists was interested in knowing if the living environment had any effect on a student's GPA. They took a set of twins and randomly assigned one twin to live in an urban area and the other twin to live in a rural area. After one year, they computed the GPAs for the twins and looked at the differences. What type of scenario is this? Matched pairs Two independent samples

Matched pairs

A university professor wanted to know if the attitudes towards statistics changed during the course of the semester. She took a simple random sample of students and gave them a test at the beginning of the term to assess their feelings toward statistics. When the semester was finished she administered another test to the same group of students and wanted to see if there was a difference between the average attitude towards statistics. What type of scenario is this? Matched pairs Two independent samples

Matched pairs

Ten percent of all customers of Cheap Foods regularly purchase Good-Enuf Brand Chicken Fingers. We plan to ask a random sample of 45 Cheap Foods customers if they regularly purchase Good-Enuf Chicken Fingers. We will then calculate hatp from the responses. Is the shape of the sampling distribution of hatp close enough to normal to use the normal distribution to compute probabilities on hatp ? Yes, because n > 30. No, because np = (45) (0.10) = 4.5 which is < 10. No, because we only have data from one sample. We cannot know the shape without knowing how many of the 45 customers purchase Good-Enuf Chicken Fingers.

No, because np = (45) (0.10) = 4.5 which is < 10.

The following histogram represents the yearly advertising budgets (in millions of dollars) of 21 randomly selected companies. A statistics student wants to create a confidence interval for the mean advertising budget of all companies. By looking at the histogram, is the use of a t procedure appropriate in this case?

No, because the data are skewed and have outliers.

A student wanted to assess the average time spent studying for his most recent exam taken in class. He asked the first 45 students who came to class how much time they spent and recorded the values. He then used this information to calculate a 95% confidence interval for the mean time spent by all students. Was this an appropriate use of the t procedure for a confidence interval?.

No, because there was no randomization in choosing the sample.

A tire manufacturer claims that one particular type of tire will last at least 50,000 miles. A group of angry customers does not believe this is so. They take a sample of 14 tires and want to test if the mean mileage of the tires is really less than 50,000. If the pvalue = 0.0200, what decision should be made if testing at the α = 0.05 level?

Reject H0 and conclude that the tires were not performing as claimed.

You may need to choose a sample size large enough to achieve a specified margin of error. However, because the sampling distribution of is a function of the population proportion p this process requires that you guess a likely value for p: p*.

The margin of error will be less than or equal to m if p* is chosen to be 0.5. Remember, though, that sample size is not always stretchable at will. There are typically costs and constraints associated with large samples.

Two-sample t-test

The null hypothesis is that both population means μ1 and μ2 are equal, thus their difference is equal to zero. H0: μ1 = μ2 <=> μ1 − μ2 = 0 with either a one-sided or a two-sided alternative hypothesis. We find how many standard errors (SE) away from (μ1 − μ2) is ( 1 − 2) by standardizing with t: Because in a two-sample test H0 poses (μ1 - μ2) = 0, we simply use Minitab will calculate the df

The confidence interval formula for p does NOT include The sample proportion. The z* value for specified level of confidence. The margin of error. The sample size. The population size.

The population size.

Significance test for p

The sampling distribution for is approximately normal for large sample sizes, and its shape depends solely on p and n. Thus, we can easily test the null hypothesis: H0: p = p0 (a given value we are testing) If H0 is true, the sampling distribution is known The likelihood of our sample proportion given the null hypothesis depends on how far from p0 our p-hat is in units of standard deviation. This is valid when both expected counts — expected successes np0 and expected failures n(1 − p0) — are each 10 or larger.

Sampling distribution of hatP

The sampling distribution of is never exactly normal. But as the sample size increases, the sampling distribution of (hatP) becomes approximately normal.

Details of the two-sample t procedures

The true value of the degrees of freedom for a two-sample t-distribution is quite lengthy to calculate. That's why some problems ask you to use an approximate value, df = smallest(n1 − 1, n2 − 1), which errs on the conservative side (often smaller than the exact). Computer software, though, gives the exact degrees of freedom — or the rounded value — for your sample data.

Robustness- two sample

The two-sample statistic is the most robust when both sample sizes are equal and both sample distributions are similar. But even when we deviate from this, two-sample tests tend to remain quite robust. As a guideline, a combined sample size (n1 + n2) of 40 or more will allow you to work even with the most skewed distributions.

The purpose of a confidence interval for p is

To give a range of reasonable values for the population proportion.

True or false: As n increases, the standard deviation of the sampling distribution of gets smaller. True False

True

The National Park Service is interested in comparing the amount of money that visitors in two different national parks spend. They sample visitors on the same day in each of the two parks and then compare the mean dollar amounts spent from each sample. What type of scenario is this? Matched pairs Two independent samples

Two independent samples

An experiment was conducted to see if elderly patients had more trouble keeping their balance when loud, unpredictable noises were made compared to younger patients who were also exposed to the noises. Researchers compared the amount of forward and backward sway for the two groups. This is an example of a Matched pairs experiment because we are analyzing the mean difference between the elderly and the young. Matched pairs experiment because we have two sets of data. Two-sample t-test because the two groups were both exposed to the noises. Two-sample t-test because the two groups are independent from one another.

Two-sample t-test because the two groups are independent from one another.

A group of psychologists was interested in knowing if the living environment had any effect on a student's GPA. They took a set of twins and randomly assigned one twin to live in an urban area and the other twin to live in a rural area. After one year, they computed the GPAs for the twins and looked at the differences. The psychologists calculated a 95% confidence interval for μd to be (-0.066, 0.146). Which one of the following shows a correct interpretation of this interval? 1. We are 95% confident that the mean GPA is somewhere in this interval. 2. 95% of the GPAs are found in that interval. 3. We are 95% confident that the mean difference between GPAs is in the interval. 4. 95% of the differences in GPAs can be found in the interval.

We are 95% confident that the mean difference between GPAs is in the interval.

Conditions for inference comparing two means

We have two independent SRSs (simple random samples) coming from two distinct populations (like men vs. women) with (μ1,σ1) and (μ2,σ2) unknown. Both populations should be Normally distributed. However, in practice, it is enough that the two distributions have similar shapes and that the sample data contain no strong outliers.

What is NOT true about the standard error of the sample mean, s/ divide n ?

We need σ when computing it.

When do we use the t-distribution to make inference about μ?

When the data are very skewed or when outliers are present. When we do not know μ or σ.

An experiment was conducted to see if elderly patients had more trouble keeping their balance when loud, unpredictable noises were made compared to younger patients who were also exposed to the noises. Researchers compared the amount of forward and backward sway for the two groups. What is the best parameter of interest when comparing the means of two groups?

u/elderly - u/ young