Which of the following statements is a characteristic of the t distribution?

When some samples are drawn from normal population whose variance is known, a distribution of the sample mean is normal. When, however, the variance of the population is unknown, the distribution is not normal but student-t, whose tail longer. That means the fact that sample mean with unknown population variance is inclined to be an extreme value. If you use normal distribution for hypothesis testing instead of t distribution, probability of error becomes bigger.

Formula :

Suppose we have a simple random sample of size n drawn from a Normal population with meanand standard deviation. Letdenote the sample mean and s, the sample standard deviation. Then the quantity

has a t distribution with n-1 degrees of freedom.

Note that there is a different t distribution for each sample size, in other words, it is a class of distributions. When we speak of a specific t distribution, we have to specify the degrees of freedom. The degrees of freedom for this t statistics comes from the sample standard deviation s in the denominator of equation 1.

The t density curves are symmetric and bell-shaped like the normal distribution and have their peak at 0. However, the spread is more than that of the standard normal distribution. This is due to the fact that in formula (1), the denominator is s rather than σ. Since s is a random quantity varying with various samples, the variability in t is more, resulting in a larger spread.

The larger the degrees of freedom, the closer the t-density is to the normal density. This reflects the fact that the standard deviation s approachesfor large sample size n. You can visualize this in the applet below by moving the sliders. 

Properties :

The Student t distribution is different for different sample sizes.

The Student t distribution is generally bell-shaped, but with smaller sample sizes shows increased variability (flatter). In other words, the distribution is less peaked than a normal distribution and with thicker tails. As the sample size increases, the distribution approaches a normal distribution. For n > 30, the differences are negligible.

The mean is zero (much like the standard normal distribution).

The distribution is symmetrical about the mean.

The variance is greater than one, but approaches one from above as the sample size increases (=1 for the standard normal distribution).

The population standard deviation is unknown.

The population is essentially normal (unimodal and basically symmetric)

Method :

   The T distribution table

Illustrations :

The current rate for producing 5 amp fuses at Neary Electric Co. is 250 per hour. A new machine has been purchased and installed that, according to the supplier, will increase the production rate. A sample of 10 randomly selected hours from the last month revealed the mean hourly production on the new machine was 256, with a sample standard deviation of 6 per hour. At the .05 significance level can Neary conclude that the new machine is faster ?

Step 1: H0: M <= 250 H1: M>250

Step 2: H0 is rejected if t>1.833, df = 9

Step 3: t = [256 - 250]/[6/sqrt(10] = 3.16

Step 4: H0 is rejected. The new machine is faster.

Applications :

It is often the case that one wants to calculate the size of sample needed to obtain a certain level of confidence in survey results. Unfortunately, this calculation requires prior knowledge of the population standard deviation (). Realistically,is unknown. Often a preliminary sample will be conducted so that a reasonable estimate of this critical population parameter can be made. If such a preliminary sample is not made, but confidence intervals for the population mean are to be constructing using an unknown, then the Student t distribution can be used.

The number of American households that were unbanked last year dropped to its lowest level since 2009, a dip due in part to people opening accounts to receive financial assistance during the pandemic, a new report says.  

Roughly 4.5% of U.S. households – or 5.9 million – didn't have a checking or savings account with a bank or credit union in 2021, a record low, according to the Federal Deposit Insurance Corporation's most recent survey of unbanked and underbanked households. 

Roughly 45% of households that received a stimulus payment, jobless benefits or other government assistance after the start of the pandemic in March, 2020 said those funds helped compel them to open an account, according to the biennial report which has been conducted since 2009.

"Safe and affordable bank accounts provide a way to bring more Americans into the banking system and will continue to play an important role in advancing economic inclusion for all Americans,'' FDIC acting chairman Martin J. Gruenberg said in a statement.  

A lack of banking options delayed some households from getting federal payments aimed at helping the country weather the economic fallout from the COVID-19 health crisis.

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The FDIC initiated an educational campaign to get more Americans to open an account to enable the direct deposit of those funds. And banks such as Capital One and Ally Financial ended  overdraft and other fees that have been a key barrier to some Americans accessing the banking system. 

What does it mean to be unbanked?

A household is deemed unbanked when no one in the home has an account with a bank or credit union. That share of households has dropped by nearly half since 2009. And since 2011, when 8% of U.S. households were unbanked, the highest since the start of the survey, and the record low reached in 2021, roughly half of the drop was due to a shift in the financial circumstances of American households the FDIC says.

Who are the underbanked?

Those who have a checking or savings account, but also use financial alternatives like check cashing services are considered underbanked. The underbanked represented 14% of U.S. households, or 18.7 million, last year.   

Why are people unbanked or underbanked?

Many of those who are unbanked say they can't afford to have an account because of the fees for insufficient funds and overdrafts that are tacked on when account balances fall short. Roughly 29% said fees or not having the required minimum balance were the primary reasons they didn't have a checking or savings account, as compared to 38% who cited those obstacles in 2019.

Are some groups more likely to be unbanked? 

The numbers of the unbanked were greater among households that included those who were working age and disabled, lower income, included a single mother, or were Black or Hispanic. Among white households for instance, 2% didn't have a bank account last year as compared to 11% and 9% of their Black and Hispanic counterparts.

Meanwhile, nearly 15% of households with a working age member who had a disability were unbanked compared to almost 4% of other households. And  nearly 16% of households with a single mother were unbanked as compared to about 2% of married couples who lacked an account. 

 "These gaps attest there's still a lot of opportunity to expand participation across the population in the banking system,'' Keith Ernst, Associate Director of Consumer Research and Examination Analytics at the FDIC, said during a media call about the report.            

Will the number of unbanked rise if the U.S. has a recession? 

Perhaps.

"During the last recession unbanked rates did indeed go up,'' Karyen Chu, chief of the Banking Research Section at the Center for Financial Research, said during the call. 

Additionally, last year, homes where the head of household was out of work were nearly five times more likely to not have a bank account as compared to those where the household head was employed.

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"To the extent that income goes down ... that has generally been associated with increases in unbanked rates,’’ Chu said. 

What are the characteristics of a t

Like the normal distribution, the t-distribution has a smooth shape. Like the normal distribution, the t-distribution is symmetric. If you think about folding it in half at the mean, each side will be the same. Like a standard normal distribution (or z-distribution), the t-distribution has a mean of zero.

What are t distributions quizlet?

t distributions. The t distributions helps specify how confident to be about research findings. The t test based on the t distributions tells us how confident we can be that the sample differs from the larger population.

What kind of distribution is the t

The t-distribution is a type of normal distribution that is used for smaller sample sizes. Normally-distributed data form a bell shape when plotted on a graph, with more observations near the mean and fewer observations in the tails.

Which of the following is not true about t distributions?

Answer and Explanation: The t distribution is symmetrical, bell-shaped and its shape depends on the degrees of freedom. The mean of the t-distribution is 0. Hence, option (B) is correct.