confidence in banks

Question

Confidence in banks: A poll conducted by the General Social Survey asked a random sample of 1325 adults in the United States how much confidence they had in banks and other financial institutions. A total of 149 adults said that they had a great deal of confidence. An economist claims that less than 15% of U.S. adults have a great deal of confidence in banks.

a.	State the appropriate null and alternate hypotheses.
b.	Compute the test statistic z.
c.	Using α = 0.05, can you conclude that the executive’s claim is true?
d.	Using α = 0.01, can you conclude that the executive’s claim is true?

Answer 1

Monica, Adding to Peter's response, the process would be as follows:

a) State the null and alternative hypothesis:

This first question is really about what are you trying to solve? In our case we are trying to see whether the proportion of people who have confidence in banks is less than 15% (using our sample as evidence to accept or reject this claim); we start by stating that 15% of the population has confidence in the bank; this means we start by stating that there is nothing new and there is no one to claim anything different (boring same!). This is the "Null Hypothesis". The "Alternative Hypothesis", i.e.some new and different information may challenge our belief) is that there is less than 15% of ppl who believe in banks (we are about to test that).

With p_hat being a sample mean, and p our population hypothesized mean:

[H(0): p_hat = p = 0.15%; and H(a): p_hat < p (i.e. p_hat < p)]

At this point ignore the sample; it simply does not yet exist.

All we need to determine is whether a sample mean p_hat (that we are about to take) is sufficiently lower than our hypothesized mean (0.15) to say that indeed we should reject that hypothesis (0.15), and state there is enough evidence to claim less than 15% of ppl have confidence in banks.

This is a one-tail to the left Z test for proportions.

Since we are testing for proportions, we can run a Z test (given sample size requirements that I will skip here); this means is we are going to use a Z table or a Z statistic to test our hypothesis.

b) Before calculating the Z statistic, let's pause and see what it means. What we said in part a) is we are trying to find out whether the sample we measure is far away enough to reject H(0); but how far away is enough?

This is where the Z statistic (and the Z critical) come into play: We will compare the calculated Z score (or Z statistic) of our sample mean to a critical Z value (based on a given criteria, i.e. alpha); this comparison will determine whether we will reject or not our null hypothesis.

The Z statistic is the number of standard deviations away from the mean; it is a numerical value that essentially is the X axis value of your standard normal distribution for a given probability. You have heard that 95% of the values fall within 2 standard deviations from the mean (on either side)? You can calculate a Z score (number of standard deviations) for any probability.

There are three ways to conduct such a test (there is absolutely no difference in the outcome and all three are valid - just a different way of saying the same thing):

1. you can determine the critical values for the population proportion p based on the Z critical values and determine whether your sample proportion falls inside or outside those limits.
2. you can compare the Z score (or Z statistic) to the Z critical value.
3. you can calculate the p-value for the test and compare it to alpha (significance level).

In this case, we are going to compare the Z statistic to a Z critical value.

Since we will want to test in c) and d) for alpha = 0.05 and alpha = 0.01, let's calculate right away our Z critical values; you can do that either from a Z table lookup or using Excel NORM.S.INV(alpha):

The Z critical value for alpha = 0.05 = -1.645

The Z critical value for alpha = 0.01 = -2.236

(again these are negative values, because this is a one-tail left test)

Now, let's calculate the Z score for our sample (we finally have a sample to work on!!).

The Z score (or Z statistic) is the number of standard deviations our sample mean sits away from the population hypothesized mean;

The number of standard deviations of the sample mean (or Standard Error) is a function of the original population proportion (NOT the sample proportion) and the sample size (n). The reason we choose the population proportion is because we know it; again here, think about not yet having taken a sample mean; you only have decided on what size sample you want to take. You will have cases where the population proportion is unknown, and you WILL need to use the sample proportion is those cases to estimate the population proportion.

The formula for the Standard Error (for proportions) is:

Standard Error = SE = √[p(1-p)/n]

With p = 0.15, and n = 1325, we get SE = 0.00981

From our sample we have p_hat = 149/1325 = 0.1125

Therefore our Z statistic = (p_hat-p)/SE = (0.1125 - 0.15)/0.00981 = -3.8276

The sample mean is therefore 3.82 standard deviations below the mean; that is far away on the left of the X axis.

c) and d)

The last pieces are easy since we have done all the work already; we just need to compare Z statistic to Z critical.

Z statistic = -3.8276

Z critical (alpha = 0.05) = -1.645

Z critical (alpha = 0.01) = -2.236

We conclude that Z statistic < Z critical (0.05) : the executive's claim is valid at alpha = 0.05 level of significance.

AND

We can also conclude that Z statistic < Z critical (0.01) : the executive's claim is also valid at alpha = 0.01 level of significance.

Interpretation:

for c) there is less than 5% chance that the sample proportion we took (0.11) is lower than the hypothesized population proportion (0.15) by chance alone. The difference is significant enough for us to reject H(0).

for d) there is less than 1% chance that the sample proportion we took (0.11) is lower than the hypothesized population proportion (0.15) by chance alone. The difference is significant enough for us to reject H(0).

Hope this helps

Answer 2

Hi Monica. We are looking for a sufficiently small p-value in order to reject the claim, Ho ,that the population proportion is 0.15 in favor of an alternative, Ha, that the population proportion is less than 0.15 based on the fact that in our survey, we generated a sample proportion, p_hat = 149/1325, based on a sample size, n, of 1325. We will use technique of Hypothesis Testing to decide what to think.

1. This is a test about a proportion, so we need to check that the conditions are met; number of observations, randomness, etc....
2. Calculate the z score for the sample proportion obtained assuming the null value of 0.15. Remember to use the standard deviation of the sampling distribution for the proportion. The z-score is the number of standard deviations that your observed value is away from the null value.
3. It's a one sided test, so we will calculate a p-value based on the probability of getting a z score less than the z score that we calculated.
4. If the p-value is less than alpha, we reject Ho, otherwise, we fail to reject it.
5. Remember we are conducting a hypothesis test. Such a test would never prove that the alternative is true, only that there is enough evidence to reject the null.