Question

The mean balance of all checking accounts at a bank on December 31, 2011, was $850. A random sample of 55 checking accounts taken recently from this bank gave a mean balance of $790 with a standard deviation of $200. Using the 1% significance level, can you conclude that the mean balance of such accounts has decreased during this period?

Answer 1

The student's question about the decrease in the mean balance of checking accounts requires a hypothesis test using the student's t-distribution, comparing the known population mean to a current sample. The method involves calculating a t-statistic and comparing it to the critical value at a 1% significance level.

The student asks whether the mean balance of checking accounts at a bank has decreased from $850 in 2011 to a current sample mean of $790, using a 1% significance level.

This question requires a hypothesis test to compare the known population mean to a sample mean using the student's t-distribution.

Given the sample size of 55 accounts and a sample standard deviation of $200, we can use a t-test to evaluate this hypothesis.

To determine the t-statistic, we use:

• t = (Sample mean - Population mean) / (Sample standard deviation / sqrt(Sample size))

Calculating the t-value and comparing it with the critical t-value from the t-distribution table at a 1% significance level will show us whether the change in mean balance is statistically significant.

If the calculated t-value exceeds the critical value, we can conclude that the mean balance has decreased significantly.

Answer 2

There is insufficient evidence to conclude that the mean balance of accounts has decreased during this period

Explanation:

We are given the following in the question:

Population mean, μ = $850

Sample mean,
`\bar{x}` = $790

Sample size, n = 55

Alpha, α = 0.01

Sample standard deviation, s = $200

First, we design the null and the alternate hypothesis

`H_(0): \mu = 850\text{ dollars}\\H_A: \mu < 850\text{ dollars}`

We use one-tailed t test to perform this hypothesis.

Formula:

`t_(stat) = \displaystyle\frac{\bar{x} - \mu}{(s)/(√(n)) }`

Putting all the values, we have

`t_(stat) = \displaystyle(790 - 850)/((200)/(√(55)) ) = -2.224`

Now,
`t_(critical) \text{ at 0.01 level of significance, 54 degree of freedom } = -2.397`

Since,

`t_(stat) > t_(critical)`

We fail to reject the null hypothesis and accept the null hypothesis. There is insufficient evidence to conclude that the mean balance of accounts has decreased during this period.